Asymptotics of solutions to the k -Yamabe equation near isolated - - PowerPoint PPT Presentation

Description of the problem Statements of Theorems Discussion of proof

Asymptotics of solutions to the σk-Yamabe equation near isolated singularities

appeared online in Invent Math, August 2010

Zheng-Chao Han, YanYan Li, Eduardo Teixeira*

Department of Mathematics, Rutgers University, Piscataway, NJ *Departamento de Matem´ atica, Universidade Federal do Cear´ a, Fortaleza-CE, Brazil.

May 30, 2012

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof

1

Description of the problem A few relevant geometric notions Various forms of the σk-Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

2

Statements of Theorems Classification of global radial solutions to the σk-Yamabe equation Main theorem Higer order expansion

3

Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Let g0 be a fixed background metric on a manifold M and g = e−2w(x)g0 be a metric conformal to g0,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Let g0 be a fixed background metric on a manifold M and g = e−2w(x)g0 be a metric conformal to g0, Ag be the Weyl-Schouten tensor of the metric g, Ag = 1 n − 2{Ric − R 2(n − 1)g} = Ag0 +

• ∇2w + dw ⊗ dw − 1

2|∇w|2g0

• .

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Let g0 be a fixed background metric on a manifold M and g = e−2w(x)g0 be a metric conformal to g0, Ag be the Weyl-Schouten tensor of the metric g, Ag = 1 n − 2{Ric − R 2(n − 1)g} = Ag0 +

• ∇2w + dw ⊗ dw − 1

2|∇w|2g0

• .

The σk curvature of g, σk(g−1 ◦ Ag) for 0 ≤ k ≤ n, is defined to be the k-th elementary symmetric function of the eigenvalues

• f the 1-1 tensor g−1 ◦ Ag.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We study the asymptotic behavior of admissible solutions w to the σk-Yamabe equation σk(g−1 ◦ Ag) = constant, (1)

• n a punctured domain BR \ {0} as x → 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We study the asymptotic behavior of admissible solutions w to the σk-Yamabe equation σk(g−1 ◦ Ag) = constant, (1)

• n a punctured domain BR \ {0} as x → 0.

An admissible solution is defined to be a C2 solution w to (1) such that for all x in its domain, Ag(x) ∈ Γ+

k , i.e.,

σj(g−1 ◦ Ag) > 0 for 1 ≤ j ≤ k.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We study the asymptotic behavior of admissible solutions w to the σk-Yamabe equation σk(g−1 ◦ Ag) = constant, (1)

• n a punctured domain BR \ {0} as x → 0.

An admissible solution is defined to be a C2 solution w to (1) such that for all x in its domain, Ag(x) ∈ Γ+

k , i.e.,

σj(g−1 ◦ Ag) > 0 for 1 ≤ j ≤ k. An admissible solution to (1) makes (1) elliptic.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Note that, since σk(g−1 ◦ Ag) = e2kwσk(g−1

• Ag),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Note that, since σk(g−1 ◦ Ag) = e2kwσk(g−1

• Ag),

(1) is equivalent to σk(g−1

• {Ag0 + ∇2w + dw ⊗ dw − 1

2|∇w|2g0}) =ce−2kw(x), (2) for some constant c.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Note that, since σk(g−1 ◦ Ag) = e2kwσk(g−1

• Ag),

(1) is equivalent to σk(g−1

• {Ag0 + ∇2w + dw ⊗ dw − 1

2|∇w|2g0}) =ce−2kw(x), (2) for some constant c. (2) is a form of the standard Yamabe equation when k = 1 and is fully nonlinear when k > 1.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We often take M = Rn and g0 to be the canonical flat metric |dx|2 in Rn.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We often take M = Rn and g0 to be the canonical flat metric |dx|2 in Rn. Writing g = u

4 n−2 (x)|dx|2, Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We often take M = Rn and g0 to be the canonical flat metric |dx|2 in Rn. Writing g = u

4 n−2 (x)|dx|2,

then equation (2) in the k = 1 case is equivalent to ∆u(x) + c′ u

n+2 n−2 (x) = 0,

and u > 0, (3) for some constant c′.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Prototype question: Suppose that u is a solution to (3) in BR \ {0}, can one characterize the behavior of u(x) as x → 0?

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Prototype question: Suppose that u is a solution to (3) in BR \ {0}, can one characterize the behavior of u(x) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c′ = 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Prototype question: Suppose that u is a solution to (3) in BR \ {0}, can one characterize the behavior of u(x) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c′ = 0. However, if u > 0 is a solution to (3) in BR \ {0} with c′ = 0, then Bocher’s theorem says that u(x) = a|x|2−n + h(x) when n ≥ 3, where h(x) is a smooth harmonic function in BR.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Prototype question: Suppose that u is a solution to (3) in BR \ {0}, can one characterize the behavior of u(x) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c′ = 0. However, if u > 0 is a solution to (3) in BR \ {0} with c′ = 0, then Bocher’s theorem says that u(x) = a|x|2−n + h(x) when n ≥ 3, where h(x) is a smooth harmonic function in BR. Note that ∆|x|2−n = 0 on Rn \ {0}.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Prototype question: Suppose that u is a solution to (3) in BR \ {0}, can one characterize the behavior of u(x) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c′ = 0. However, if u > 0 is a solution to (3) in BR \ {0} with c′ = 0, then Bocher’s theorem says that u(x) = a|x|2−n + h(x) when n ≥ 3, where h(x) is a smooth harmonic function in BR. Note that ∆|x|2−n = 0 on Rn \ {0}. The same question for the c′ > 0 case was resolved in the celebrated work of Caffarelli, Gidas, and Spruck, and will be reviewed later.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln |x| and θ = x/|x| (so that |dx|2 = |x|2(dt2 + dθ2)),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln |x| and θ = x/|x| (so that |dx|2 = |x|2(dt2 + dθ2)), and new variables U(t, θ) and w(t, θ) by u

4 n−2 (x)|dx|2 = U 4 n−2 (t, θ)(dt2 + dθ2)

= e−2w(t,θ)(dt2 + dθ2).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln |x| and θ = x/|x| (so that |dx|2 = |x|2(dt2 + dθ2)), and new variables U(t, θ) and w(t, θ) by u

4 n−2 (x)|dx|2 = U 4 n−2 (t, θ)(dt2 + dθ2)

= e−2w(t,θ)(dt2 + dθ2). Thus |x|

n−2 2 u(x) = U(t, θ) = e− n−2 2 w(t,θ). Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln |x| and θ = x/|x| (so that |dx|2 = |x|2(dt2 + dθ2)), and new variables U(t, θ) and w(t, θ) by u

4 n−2 (x)|dx|2 = U 4 n−2 (t, θ)(dt2 + dθ2)

= e−2w(t,θ)(dt2 + dθ2). Thus |x|

n−2 2 u(x) = U(t, θ) = e− n−2 2 w(t,θ).

Written in terms of U(t, θ) and with c′ = n(n − 2)/4, (3) becomes Utt(t, θ)+∆Sn−1U(t, θ)−(n − 2)2 4 U(t, θ)+n(n − 2) 4 U

n+2 n−2 (t, θ) = 0.

(4)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

When w(t, θ) = ξ(t) is a radial solution to (2), written with respect to the background metric dt2 + dθ2 and c normalized to be 2−kn

k

• ,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

When w(t, θ) = ξ(t) is a radial solution to (2), written with respect to the background metric dt2 + dθ2 and c normalized to be 2−kn

k

• , (2) reduces to the following ODE

2(1 − ξ2

t )k−1

k nξtt + (1 2 − k n)(1 − ξ2

t )

• e2kξ = 1.

(5)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

When w(t, θ) = ξ(t) is a radial solution to (2), written with respect to the background metric dt2 + dθ2 and c normalized to be 2−kn

k

• , (2) reduces to the following ODE

2(1 − ξ2

t )k−1

k nξtt + (1 2 − k n)(1 − ξ2

t )

• e2kξ = 1.

(5) (5) is not fully nonlinear, but may become degenerate in the case k > 1 when 1 − ξ2

t → 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

For a metric g whose Schouten-Weyl tensor is in the Γ+

k class

with k > 1, we know (i) σk(g−1 ◦ Ag) places a stronger control on the curvature tensor than the scalar curvature: Chang, Gursky and Yang [CGY1] observed that if σ1(g−1 ◦ Ag), σ2(g−1 ◦ Ag) > 0 at a point on a 4-dimensional manifold, then the Ricci tensor of g is positive definite at that point;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

For a metric g whose Schouten-Weyl tensor is in the Γ+

k class

with k > 1, we know (i) σk(g−1 ◦ Ag) places a stronger control on the curvature tensor than the scalar curvature: Chang, Gursky and Yang [CGY1] observed that if σ1(g−1 ◦ Ag), σ2(g−1 ◦ Ag) > 0 at a point on a 4-dimensional manifold, then the Ricci tensor of g is positive definite at that point; this algebraic relation has been generalized to higher dimensions by Guan, Viaclovsky, and Wang [GVW]; and (ii) the expression σk(g−1 ◦ Ag) is a fully nonlinear PDO in w that becomes elliptic.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Singular sets and the asymptotics of singular solutions to equation (2) are often intimately related to the underlying conformal geometry of the manifold.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Singular sets and the asymptotics of singular solutions to equation (2) are often intimately related to the underlying conformal geometry of the manifold. Loewner and Nirenberg studied solutions to (3) on Rn \ Γ with c = −1, where Γ stands for the singular set for u.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Singular sets and the asymptotics of singular solutions to equation (2) are often intimately related to the underlying conformal geometry of the manifold. Loewner and Nirenberg studied solutions to (3) on Rn \ Γ with c = −1, where Γ stands for the singular set for u. They showed that if Γ is a closed set with Hausdorff dimension < (n − 2)/2, then any solution u to (3) on Rn \ Γ with c = −1 is in fact smooth across Γ,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

and that if Γ is a closed smooth submanifold of Rn with dimension > (n − 2)/2, then there is a solution to (3) on Rn \ Γ with c = −1 which has non-removable singularity

• ver Γ.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

and that if Γ is a closed smooth submanifold of Rn with dimension > (n − 2)/2, then there is a solution to (3) on Rn \ Γ with c = −1 which has non-removable singularity

• ver Γ. Later improvements and generalizations were
• btained by Aviles, Veron, McOwen, D. Finn, and others.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

and that if Γ is a closed smooth submanifold of Rn with dimension > (n − 2)/2, then there is a solution to (3) on Rn \ Γ with c = −1 which has non-removable singularity

• ver Γ. Later improvements and generalizations were
• btained by Aviles, Veron, McOwen, D. Finn, and others.

Schoen and Yau studied solutions to (3) on Rn \ Γ with c = 1 in connection with their study of developing maps of locally conformally flat manifolds.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

and that if Γ is a closed smooth submanifold of Rn with dimension > (n − 2)/2, then there is a solution to (3) on Rn \ Γ with c = −1 which has non-removable singularity

• ver Γ. Later improvements and generalizations were
• btained by Aviles, Veron, McOwen, D. Finn, and others.

Schoen and Yau studied solutions to (3) on Rn \ Γ with c = 1 in connection with their study of developing maps of locally conformally flat manifolds. Their results imply that if u is a solution to (3) on Rn \ Γ with c = 1 such that the corresponding conformal metric g is complete, then the Hausdorff dimension of Γ is ≤ (n − 2)/2.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Schoen also constructed solutions to (3) on Rn \ Γ with c = 1 that are singular at Γ, where Γ is any prescribed finite set of more than one point.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Schoen also constructed solutions to (3) on Rn \ Γ with c = 1 that are singular at Γ, where Γ is any prescribed finite set of more than one point. Constructions of solutions with more general prescribed singular sets, subject to the dimensional constraints, were later obtained by N. Smale,

• R. Mazzeo, F

. Pacard, and others.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Schoen also constructed solutions to (3) on Rn \ Γ with c = 1 that are singular at Γ, where Γ is any prescribed finite set of more than one point. Constructions of solutions with more general prescribed singular sets, subject to the dimensional constraints, were later obtained by N. Smale,

• R. Mazzeo, F

. Pacard, and others. Chang, Hang, and Yang proved that if Ω ⊂ Sn (n ≥ 5) admits a complete, conformal metric g σ1(Ag) ≥ c1 > 0, σ2(Ag) ≥ 0, and |Rg| + |∇gR|g ≤ c0, (6) then dim(Sn \ Ω) < (n − 4)/2.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

This has been generalized by M. Gonzalez to the case of 2 < k < n/2: if Ω ⊂ Sn admits a complete, conformal metric g with σ1(Ag) ≥ c1 > 0, σ2(Ag), · · · , σk(Ag) ≥ 0, and (6), then dim(Sn \ Ω) < (n − 2k)/2.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Global solutions to (3) on Rn, or on Rn with one point deleted, were classified by Caffarelli, Gidas, and Spruck (earlier, related results were due to Obata, and Gidas-Ni-Nirenberg).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Global solutions to (3) on Rn, or on Rn with one point deleted, were classified by Caffarelli, Gidas, and Spruck (earlier, related results were due to Obata, and Gidas-Ni-Nirenberg). Furthermore, they obtained asymptotics for solutions to (3) with isolated singularity.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Global solutions to (3) on Rn, or on Rn with one point deleted, were classified by Caffarelli, Gidas, and Spruck (earlier, related results were due to Obata, and Gidas-Ni-Nirenberg). Furthermore, they obtained asymptotics for solutions to (3) with isolated singularity. More precisely, they proved Theorem A Suppose that u(x) is a positive solution to (3) in BR \ {0} and does not extend to a smooth solution to (3) over 0, then u(x) = ¯ u(|x|) (1 + O(|x|)) as x → 0, (7) with ¯ u(|x|) = \

• u(|x|θ) dθ

being the spherical average of u over the sphere ∂B (0);

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

furthermore, there exists a radial singular solution u∗(|x|) to (3)

• n Rn \ {0} and some α > 0, 0 < ǫ ≤ ǫ0 and τ such that

u(x) = u∗(|x|) (1 + O(|x|α)) as |x| → 0. (8) and u∗(|x|) = |x|− n−2

2 ψǫ(− ln |x| + τ)

as |x| → 0, where ψǫ(t) is an entire radial solution to (4) on R.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

furthermore, there exists a radial singular solution u∗(|x|) to (3)

• n Rn \ {0} and some α > 0, 0 < ǫ ≤ ǫ0 and τ such that

u(x) = u∗(|x|) (1 + O(|x|α)) as |x| → 0. (8) and u∗(|x|) = |x|− n−2

2 ψǫ(− ln |x| + τ)

as |x| → 0, where ψǫ(t) is an entire radial solution to (4) on R. Alternative approach and refined asymptotics for the same problem were later obtained by Korevaar, Mazzeo, Pacard and Schoen.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

furthermore, there exists a radial singular solution u∗(|x|) to (3)

• n Rn \ {0} and some α > 0, 0 < ǫ ≤ ǫ0 and τ such that

u(x) = u∗(|x|) (1 + O(|x|α)) as |x| → 0. (8) and u∗(|x|) = |x|− n−2

2 ψǫ(− ln |x| + τ)

as |x| → 0, where ψǫ(t) is an entire radial solution to (4) on R. Alternative approach and refined asymptotics for the same problem were later obtained by Korevaar, Mazzeo, Pacard and Schoen. There have also been many other papers related to the themes of Theorem A, including a recent work of Marques dealing with background metric not necessarily locally conformally flat in low dimensions.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities. Other related questions are: Given a subset S of Sn, contruct (admissible) solutions of (2) on Sn \ S which are singulalr on S and gives rise to a complete metric on Sn \ S.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities. Other related questions are: Given a subset S of Sn, contruct (admissible) solutions of (2) on Sn \ S which are singulalr on S and gives rise to a complete metric on Sn \ S. Given a subset S of Sn, study the moduli space of (admissible) solutions of (2) on Sn \ S which are singulalr

• n S and gives rise to a complete metric on Sn \ S.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof A few relevant geometric notions Various forms of the σk -Yamabe equation Why study solutions in Γ+

k class?

Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case

Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities. Other related questions are: Given a subset S of Sn, contruct (admissible) solutions of (2) on Sn \ S which are singulalr on S and gives rise to a complete metric on Sn \ S. Given a subset S of Sn, study the moduli space of (admissible) solutions of (2) on Sn \ S which are singulalr

• n S and gives rise to a complete metric on Sn \ S.
• L. Mazzieri, C. B. Ndiaye and A. Segatti have obtained some

results to the first question above.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

First, we record part of the results by Chang, Han and Yang on the classification of global radial solutions to (2) that will be useful in our settings.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

First, we record part of the results by Chang, Han and Yang on the classification of global radial solutions to (2) that will be useful in our settings. In very rough terms, w(t) = ln cosh(t) is always a solution to (2), which corresponds to u4/(n−2)(x)|dx|2 being the round spherical metric

• 2

1 + |x|2 2 |dx|2.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

First, we record part of the results by Chang, Han and Yang on the classification of global radial solutions to (2) that will be useful in our settings. In very rough terms, w(t) = ln cosh(t) is always a solution to (2), which corresponds to u4/(n−2)(x)|dx|2 being the round spherical metric

• 2

1 + |x|2 2 |dx|2. when 2k < n, there exists a two parameter family of undulating solutions (called Delaunay type solutions in

• ther contexts), periodic in t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

When 2k > n, all radial solutions defined in BR \ {0} have the form u(x) = u0 + A|x|2−n/k + h.o.t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

When 2k > n, all radial solutions defined in BR \ {0} have the form u(x) = u0 + A|x|2−n/k + h.o.t. When 2k = n, all radial solutions defined in BR \ {0}, in terms of U, have the form U = U0e

− n−2

2

√ 1− k √ h

• t

+ h.o.t, for some 0 ≤ h < 1, as |x| = e−t → 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

For any solution ξ(t) of (5), e(2k−n)ξ(t)(1 − ξ2

t (t))k − e−nξ(t)

is a constant, which we call the first integral of the solution ξ(t).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

For any solution ξ(t) of (5), e(2k−n)ξ(t)(1 − ξ2

t (t))k − e−nξ(t)

is a constant, which we call the first integral of the solution ξ(t). We can parametrize the global singular radial solutions to (5) as follows: for each 0 < h, subject to any further constraints depending on 2k < or = n, let ξh(t) denote the solution to (5) with its first integral equal to h and such that ξh(0) equals minR ξh(t).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

For any solution ξ(t) of (5), e(2k−n)ξ(t)(1 − ξ2

t (t))k − e−nξ(t)

is a constant, which we call the first integral of the solution ξ(t). We can parametrize the global singular radial solutions to (5) as follows: for each 0 < h, subject to any further constraints depending on 2k < or = n, let ξh(t) denote the solution to (5) with its first integral equal to h and such that ξh(0) equals minR ξh(t). Next we state our main result.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

Theorem 1 Let w(t, θ) be a solution to (2) on {t > t0} × Sn−1 in the Γ+

k

class, where the constant is normalized to be 2−kn

k

• . Then

there exists α > 0, h and τ such that |w(t, θ) − ξh(t + τ)| ≤ Ce−αt. (9)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

Theorem 1 Let w(t, θ) be a solution to (2) on {t > t0} × Sn−1 in the Γ+

k

class, where the constant is normalized to be 2−kn

k

• . Then

there exists α > 0, h and τ such that |w(t, θ) − ξh(t + τ)| ≤ Ce−αt. (9) Remarks In terms of the variable u(x) defined on BR \ {0}, if lim inf

x→0 |x|

n−2 2 u(x) > 0,

(10) then 2k < n and 0 < h ≤ h∗, furthermore

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

u(x) = (1 + o(|x|α)) |x|− n−2

2 e− n−2 2 ξh(− ln |x|+τ)

(11) as x → 0;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

u(x) = (1 + o(|x|α)) |x|− n−2

2 e− n−2 2 ξh(− ln |x|+τ)

(11) as x → 0; if 2k > n, or 2k < n and lim inf

x→0 |x|

n−2 2 u(x) = 0,

(12) then limx→0 u(x) exists and equals some a > 0, and there exist some α > 0 and C > 0 such that |u(x) − a| ≤ C|x|α; (13)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

if 2k = n, then 0 ≤ h < 1 and for some α > 0, |x|

n−2 2 (1−

√

1− k √ h)u(x)

extends to a Cα positive function over BR.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

if 2k = n, then 0 ≤ h < 1 and for some α > 0, |x|

n−2 2 (1−

√

1− k √ h)u(x)

extends to a Cα positive function over BR. Remarks In the case 2k > n Gursky and Viaclovsky, YanYan Li had

• btained (13) earlier. In the case 2k < n, M. Gonzalez proved

that if u is a solution to (2) in BR \ {0} in the Γ+

k class such that

u4/(n−2)|dx|2 has finite volume over BR \ {0} , then u is bounded in BR \ {0}.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2k ≤ n. Theorem 2 Let w(t, θ) be a solution to (2) on {t > t0} × Sn−1 in the Γ+

k

class, where n ≥ 3, 2 ≤ k ≤ n/2, and the constant c is normalized to be 2−kn

k

• ,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2k ≤ n. Theorem 2 Let w(t, θ) be a solution to (2) on {t > t0} × Sn−1 in the Γ+

k

class, where n ≥ 3, 2 ≤ k ≤ n/2, and the constant c is normalized to be 2−kn

k

• , and let w∗(t) = ξh(t + τ) be the radial

solution to (2) on R × Sn−1 in the Γ+

k class for which (9) holds.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2k ≤ n. Theorem 2 Let w(t, θ) be a solution to (2) on {t > t0} × Sn−1 in the Γ+

k

class, where n ≥ 3, 2 ≤ k ≤ n/2, and the constant c is normalized to be 2−kn

k

• , and let w∗(t) = ξh(t + τ) be the radial

solution to (2) on R × Sn−1 in the Γ+

k class for which (9) holds.

Let {Yj(θ) : j = 0, 1, · · · } denote the set of normalized spherical harmonics,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2k ≤ n. Theorem 2 Let w(t, θ) be a solution to (2) on {t > t0} × Sn−1 in the Γ+

k

class, where n ≥ 3, 2 ≤ k ≤ n/2, and the constant c is normalized to be 2−kn

k

• , and let w∗(t) = ξh(t + τ) be the radial

solution to (2) on R × Sn−1 in the Γ+

k class for which (9) holds.

Let {Yj(θ) : j = 0, 1, · · · } denote the set of normalized spherical harmonics, and ρ be the infimum of the positive characteristic exponents to the linearized equation of (2) at w∗(t) corresponding to higher order spherical harmonics Yj(θ), j > n

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

Then ρ > 1, and there is a w1(t, θ) =

n

• j=1

aje−t−τ 1 + ξ

′

h(t + τ)

• Yj(θ),

which is a solution to the linearized equation of (2) at w∗(t),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

Then ρ > 1, and there is a w1(t, θ) =

n

• j=1

aje−t−τ 1 + ξ

′

h(t + τ)

• Yj(θ),

which is a solution to the linearized equation of (2) at w∗(t), such that |w(t, θ) − w∗(t) − w1(t, θ)| ≤ Ce−min{2,ρ}t for t > t0 + 1, (14) provided ρ = 2;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

Then ρ > 1, and there is a w1(t, θ) =

n

• j=1

aje−t−τ 1 + ξ

′

h(t + τ)

• Yj(θ),

which is a solution to the linearized equation of (2) at w∗(t), such that |w(t, θ) − w∗(t) − w1(t, θ)| ≤ Ce−min{2,ρ}t for t > t0 + 1, (14) provided ρ = 2; when ρ = 2, (14) continues to hold if the right hand side is modified into Cte−2t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Classification of global radial solutions to the σk -Yamabe equation Main theorem Higer order expansion

Then ρ > 1, and there is a w1(t, θ) =

n

• j=1

aje−t−τ 1 + ξ

′

h(t + τ)

• Yj(θ),

which is a solution to the linearized equation of (2) at w∗(t), such that |w(t, θ) − w∗(t) − w1(t, θ)| ≤ Ce−min{2,ρ}t for t > t0 + 1, (14) provided ρ = 2; when ρ = 2, (14) continues to hold if the right hand side is modified into Cte−2t. Theorem 2 requires some knowledge on the spectrum of the linearized operator to (2).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We are able to provide the needed analysis. Such analysis will also be needed in constructing solutions to (2) on Sn \ Λ, and in analysing the moduli space of solutions to (2) on Sn \ Λ, when Λ is a finite set.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We are able to provide the needed analysis. Such analysis will also be needed in constructing solutions to (2) on Sn \ Λ, and in analysing the moduli space of solutions to (2) on Sn \ Λ, when Λ is a finite set. Our knowledge of the spectrum of the linearized operator to (2) immediately yields Fredholm mapping properties of these

• perators on appropriately defined weighted spaces, as those

in [MPU], [MS], and [KMPS].

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We are able to provide the needed analysis. Such analysis will also be needed in constructing solutions to (2) on Sn \ Λ, and in analysing the moduli space of solutions to (2) on Sn \ Λ, when Λ is a finite set. Our knowledge of the spectrum of the linearized operator to (2) immediately yields Fredholm mapping properties of these

• perators on appropriately defined weighted spaces, as those

in [MPU], [MS], and [KMPS]. We are able to adapt either of the approach in [CGS] or [KMPS] to prove the main part of Theorem 1.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The approach in [CGS] first proves that w(t, θ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3);

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The approach in [CGS] first proves that w(t, θ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3); while the approach in [KMPS] makes use of rescaling technique to extract subsequential convergence (on compact domains) to singular global radial solutions,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The approach in [CGS] first proves that w(t, θ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3); while the approach in [KMPS] makes use of rescaling technique to extract subsequential convergence (on compact domains) to singular global radial solutions, and makes essential use of the spectrum of the linearized operator to (2) at the singular global radial solution to (2).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The approach in [CGS] first proves that w(t, θ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3); while the approach in [KMPS] makes use of rescaling technique to extract subsequential convergence (on compact domains) to singular global radial solutions, and makes essential use of the spectrum of the linearized operator to (2) at the singular global radial solution to (2). Both approaches need a Liouville type classification results which show that all global singular solution to (2) are radial.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The needed classification results are obtained by combining a Liouville type result of Y.Y Li for solutions to equations that include (2) and the classification results of radial solutions to (2).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The needed classification results are obtained by combining a Liouville type result of Y.Y Li for solutions to equations that include (2) and the classification results of radial solutions to (2). Proposition 1 Let U(t, θ) = e− n−2

2 w(t,θ)

be any positive solution to (2) defined on the entire cylinder R × Sn−1. Suppose that U

4 n−2 (t, θ)(dt2 + dθ2) is in the Γ+

k

class.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The needed classification results are obtained by combining a Liouville type result of Y.Y Li for solutions to equations that include (2) and the classification results of radial solutions to (2). Proposition 1 Let U(t, θ) = e− n−2

2 w(t,θ)

be any positive solution to (2) defined on the entire cylinder R × Sn−1. Suppose that U

4 n−2 (t, θ)(dt2 + dθ2) is in the Γ+

k

class.Then either u(x) = |x|− n−2

2 U(− ln |x|, x

|x|)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

can be extended as a C2 positive function near 0, in which case u(x) is a radially symmetric solution to (2) about some point in Rn;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

can be extended as a C2 positive function near 0, in which case u(x) is a radially symmetric solution to (2) about some point in Rn; or u(x) can’t be extended as a C2 positive function near 0, and U is independent of θ.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

can be extended as a C2 positive function near 0, in which case u(x) is a radially symmetric solution to (2) about some point in Rn; or u(x) can’t be extended as a C2 positive function near 0, and U is independent of θ. Moreover, when 2k < n, U(t) is a periodic solution of (1) with 0 < U(t) ≤ 1 for all t ∈ R and the first integral h > 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

can be extended as a C2 positive function near 0, in which case u(x) is a radially symmetric solution to (2) about some point in Rn; or u(x) can’t be extended as a C2 positive function near 0, and U is independent of θ. Moreover, when 2k < n, U(t) is a periodic solution of (1) with 0 < U(t) ≤ 1 for all t ∈ R and the first integral h > 0. We refer to these solutions as global singular solutions to (1).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Let me first outline a few key steps in our proof using the spectrum analysis for the case 2k < n.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Let me first outline a few key steps in our proof using the spectrum analysis for the case 2k < n. The asymptotic behavior of u(x) as x → 0 is encoded in the asymptotic behavior of w(t, θ) as t → ∞, which we will study through the limit(s) of {w(t + tj, θ)} for any sequence tj → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Let me first outline a few key steps in our proof using the spectrum analysis for the case 2k < n. The asymptotic behavior of u(x) as x → 0 is encoded in the asymptotic behavior of w(t, θ) as t → ∞, which we will study through the limit(s) of {w(t + tj, θ)} for any sequence tj → ∞. Step 1. Get an L∞ bound for w(t, θ), so that the sequence {w(t + tj, θ)} is bounded for any sequence tj → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Let me first outline a few key steps in our proof using the spectrum analysis for the case 2k < n. The asymptotic behavior of u(x) as x → 0 is encoded in the asymptotic behavior of w(t, θ) as t → ∞, which we will study through the limit(s) of {w(t + tj, θ)} for any sequence tj → ∞. Step 1. Get an L∞ bound for w(t, θ), so that the sequence {w(t + tj, θ)} is bounded for any sequence tj → ∞. Once an L∞ bound for w(t, θ) is obtained, we will aim for gradient estimates of w(t, θ).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Let me first outline a few key steps in our proof using the spectrum analysis for the case 2k < n. The asymptotic behavior of u(x) as x → 0 is encoded in the asymptotic behavior of w(t, θ) as t → ∞, which we will study through the limit(s) of {w(t + tj, θ)} for any sequence tj → ∞. Step 1. Get an L∞ bound for w(t, θ), so that the sequence {w(t + tj, θ)} is bounded for any sequence tj → ∞. Once an L∞ bound for w(t, θ) is obtained, we will aim for gradient estimates of w(t, θ). In fact, for solution w(t, θ) to (2), there is local gradient estimates of w(t, θ) in terms of an upper bound for e−w(t,θ) through the work of Guan-Wang.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

An upper bound for e−w(t,θ) is obtained throught the following result of Y.Y. Li. Proposition 2 Suppose that u ∈ C2(B2 \ {0}) is a positive function such that g = u

4 n−2 (x)|dx|2

is a metric in Γ+

k and gives rise to a solution to (1) over B2 \ {0}.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

An upper bound for e−w(t,θ) is obtained throught the following result of Y.Y. Li. Proposition 2 Suppose that u ∈ C2(B2 \ {0}) is a positive function such that g = u

4 n−2 (x)|dx|2

is a metric in Γ+

k and gives rise to a solution to (1) over B2 \ {0}.

Then lim sup

x→0

|x|

n−2 2 u(x) < ∞,

(15)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

and there exists some constant C > 0 such that |u(x) − ¯ u(|x|)| ≤ C|x|¯ u(|x|), (16) for 0 < |x| ≤ 1, where ¯ u(|x|) = 1 |∂B|x|(0)|

• ∂B|x|(0)

u(y)dσ(y) is the spherical average of u(x) over ∂B|x|(0).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In terms of the variables w(t, θ) = − 2 n − 2 ln U(t, θ) = − 2 n − 2u(e−tθ) + t, and γ(t) := |Sn−1|−1

• Sn−1 w(t, θ)d θ,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In terms of the variables w(t, θ) = − 2 n − 2 ln U(t, θ) = − 2 n − 2u(e−tθ) + t, and γ(t) := |Sn−1|−1

• Sn−1 w(t, θ)d θ,

(15) implies that e−2w(t,θ) ≤ C1, (17) for some constant C1,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In terms of the variables w(t, θ) = − 2 n − 2 ln U(t, θ) = − 2 n − 2u(e−tθ) + t, and γ(t) := |Sn−1|−1

• Sn−1 w(t, θ)d θ,

(15) implies that e−2w(t,θ) ≤ C1, (17) for some constant C1, and (16) implies that,

• w(t, θ) := w(t, θ) − γ(t) satisfies

| w(t, θ)| ≤ C2e−t, (18) for some constant C2.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Using (18), gradients estimates and interpolation, we can

• btain, for any 1 > δ > 0, a constant C > 0 such that

|∇l

t,θ(w(t, θ) − γ(t))| ≤ Ce−(1−δ)t,

(19) for all t ≥ 0 and 1 ≤ l ≤ 2.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Using (18), gradients estimates and interpolation, we can

• btain, for any 1 > δ > 0, a constant C > 0 such that

|∇l

t,θ(w(t, θ) − γ(t))| ≤ Ce−(1−δ)t,

(19) for all t ≥ 0 and 1 ≤ l ≤ 2. Step 2. Make use of a Pohozaev type identity for solutions w(t, θ) to (2), which takes the form

• Sn−1
• ne(2k−n)w(t,θ)

2kσk

n

• a=1

Ta1[w(t, θ)]wat(t, θ) − e−nw(t,θ)

• dθ =

h (20) for some constant h independent of t,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Using (18), gradients estimates and interpolation, we can

• btain, for any 1 > δ > 0, a constant C > 0 such that

|∇l

t,θ(w(t, θ) − γ(t))| ≤ Ce−(1−δ)t,

(19) for all t ≥ 0 and 1 ≤ l ≤ 2. Step 2. Make use of a Pohozaev type identity for solutions w(t, θ) to (2), which takes the form

• Sn−1
• ne(2k−n)w(t,θ)

2kσk

n

• a=1

Ta1[w(t, θ)]wat(t, θ) − e−nw(t,θ)

• dθ =

h (20) for some constant h independent of t, where Ta1[w(t, θ)] are the components of the Newton tensor associated with σk(Aw(t,θ)).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(20) can be expressed in terms of γ(t) via the help of estimate (19), as e(2k−n)γ (1 − γ2

t )k + η3(t)

• − e−nγ {1 + η4(t)} = h,

(21) where h ≥ 0 is a constant multiple of h, ηi(t), for i = 3, 4, have the decay rate ηi(t) = O(e−2(1−δ)t) as t → ∞, and δ > 0 can be made as small as one needs.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(20) can be expressed in terms of γ(t) via the help of estimate (19), as e(2k−n)γ (1 − γ2

t )k + η3(t)

• − e−nγ {1 + η4(t)} = h,

(21) where h ≥ 0 is a constant multiple of h, ηi(t), for i = 3, 4, have the decay rate ηi(t) = O(e−2(1−δ)t) as t → ∞, and δ > 0 can be made as small as one needs. h can be thought of as a numerical invariant of the singular behavior of u(x) near 0, and is equal to 0 when u(x) is smooth

• ver 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 3. Identifying the limits of {w(t + tj, θ)}. When h > 0 and 2k < n, (21) and the gradient estimates imply that γ(t), therefore, w(t, θ) is bounded from above.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 3. Identifying the limits of {w(t + tj, θ)}. When h > 0 and 2k < n, (21) and the gradient estimates imply that γ(t), therefore, w(t, θ) is bounded from above. Then it follows that for any sequence tj → ∞, a subsequence of {w(t + tj, θ)} converges to some bounded admissible solution w∞(t, θ) to (2) defined on the entire cylinder R × Sn−1.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 3. Identifying the limits of {w(t + tj, θ)}. When h > 0 and 2k < n, (21) and the gradient estimates imply that γ(t), therefore, w(t, θ) is bounded from above. Then it follows that for any sequence tj → ∞, a subsequence of {w(t + tj, θ)} converges to some bounded admissible solution w∞(t, θ) to (2) defined on the entire cylinder R × Sn−1. The classification results now say that w∞(t, θ) = ξh(t + τ) for some τ, where h is the same as in (21),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 3. Identifying the limits of {w(t + tj, θ)}. When h > 0 and 2k < n, (21) and the gradient estimates imply that γ(t), therefore, w(t, θ) is bounded from above. Then it follows that for any sequence tj → ∞, a subsequence of {w(t + tj, θ)} converges to some bounded admissible solution w∞(t, θ) to (2) defined on the entire cylinder R × Sn−1. The classification results now say that w∞(t, θ) = ξh(t + τ) for some τ, where h is the same as in (21), but τ may depend on the choice of the sequence tj → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 3. Identifying the limits of {w(t + tj, θ)}. When h > 0 and 2k < n, (21) and the gradient estimates imply that γ(t), therefore, w(t, θ) is bounded from above. Then it follows that for any sequence tj → ∞, a subsequence of {w(t + tj, θ)} converges to some bounded admissible solution w∞(t, θ) to (2) defined on the entire cylinder R × Sn−1. The classification results now say that w∞(t, θ) = ξh(t + τ) for some τ, where h is the same as in (21), but τ may depend on the choice of the sequence tj → ∞. Our remaining job is to show that τ is indepedndent of the choice of the sequence tj → ∞, and in fact, w(t, θ) − ξh(t + τ) → 0 at an exponential rate as t → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 4. Unique limit and exponential convergence.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 4. Unique limit and exponential convergence. We first summarize the covergence behavior of w(t, θ) as t → ∞— these correspond to the approach in [KMPS]. (a) Let tj → ∞ be any sequence tending to ∞, then {wj(t, θ) := w(t + tj, θ)} has a subsequence converging to a bounded limiting solution ξ(t) of (3) defined for (t, θ) ∈ R × Sn−1. The convergence is uniform on any compact subset of R+ × Sn−1.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 4. Unique limit and exponential convergence. We first summarize the covergence behavior of w(t, θ) as t → ∞— these correspond to the approach in [KMPS]. (a) Let tj → ∞ be any sequence tending to ∞, then {wj(t, θ) := w(t + tj, θ)} has a subsequence converging to a bounded limiting solution ξ(t) of (3) defined for (t, θ) ∈ R × Sn−1. The convergence is uniform on any compact subset of R+ × Sn−1. (b) Any angular derivative ∂θw(t, θ) of w converges to 0 as t → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Step 4. Unique limit and exponential convergence. We first summarize the covergence behavior of w(t, θ) as t → ∞— these correspond to the approach in [KMPS]. (a) Let tj → ∞ be any sequence tending to ∞, then {wj(t, θ) := w(t + tj, θ)} has a subsequence converging to a bounded limiting solution ξ(t) of (3) defined for (t, θ) ∈ R × Sn−1. The convergence is uniform on any compact subset of R+ × Sn−1. (b) Any angular derivative ∂θw(t, θ) of w converges to 0 as t → ∞. (c) There exists S > 0 such that for any infinitesimal rotation ∂θ of Sn−1, and for any tj → ∞, if we set Aj = supt≥0 |∂θwj(t, θ)|, and if |∂θwj(sj, θj)| = Aj for some (sj, θj) ∈ R+ × Sn−1, then sj ≤ S.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(d) ∂θw(t, θ) converges to 0 at an exponential rate as t → ∞, and |w(t, θ) − |Sn−1|−1

• Sn−1 w(t, ω)dω|

converges to 0 at an exponential rate as t → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(d) ∂θw(t, θ) converges to 0 at an exponential rate as t → ∞, and |w(t, θ) − |Sn−1|−1

• Sn−1 w(t, ω)dω|

converges to 0 at an exponential rate as t → ∞. (e) There exists a bounded (periodic) solution ξ(t) of (3) and τ such that w(t, θ) converges to ξ(t + τ) at an exponential rate as t → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(d) ∂θw(t, θ) converges to 0 at an exponential rate as t → ∞, and |w(t, θ) − |Sn−1|−1

• Sn−1 w(t, ω)dω|

converges to 0 at an exponential rate as t → ∞. (e) There exists a bounded (periodic) solution ξ(t) of (3) and τ such that w(t, θ) converges to ξ(t + τ) at an exponential rate as t → ∞. (a)–(e) are proved along almost identical lines as in the proof for Proposition 5 in [KMPS], provided some analytical preparations on the spectrum of the linearized operator of (2) at a radial solution ξ(t) are established.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The linearization of σk(Ag) at g = e−2ξ(t)(dt2 + dθ2) is Lξ(φ) =(1 − |ξt(t)|2)k−2 2k−2 n − 1 k − 1

• [A(t)φtt(t, θ) + B(t)φt(t, θ)

+C(t)∆θφ(t, θ)] ,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

The linearization of σk(Ag) at g = e−2ξ(t)(dt2 + dθ2) is Lξ(φ) =(1 − |ξt(t)|2)k−2 2k−2 n − 1 k − 1

• [A(t)φtt(t, θ) + B(t)φt(t, θ)

+C(t)∆θφ(t, θ)] , where A(t) = (1 − |ξt(t)|2) 2 , (22) B(t) = −ξt(t)

• (k − 1)ξtt(t) + n − 2k

2 (1 − |ξt(t)|2)

• ,

(23) C(t) = k − 1 n − 1ξtt(t) + n − 2k + 1 n − 1 · 1 − |ξt(t)|2 2 . (24)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

When ξ(t) is a solution to σk(g−1 ◦ Ag) = const., normalized to be 2−kn

k

• , the linearization of the fully nonlinear PDE (2) at ξ(t)

is then Lξ(φ) + 21−kk n k

• e−2kξ(t)φ(t, θ) = 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

When ξ(t) is a solution to σk(g−1 ◦ Ag) = const., normalized to be 2−kn

k

• , the linearization of the fully nonlinear PDE (2) at ξ(t)

is then Lξ(φ) + 21−kk n k

• e−2kξ(t)φ(t, θ) = 0.

If we take the projections of φ(t, ·) into spherical harmonics: φ(t, θ) =

• j

φj(t)Yj(θ), where Yj(θ) are the normalized eigenfunctions of ∆θ on L2(Sn−1).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

When ξ(t) is a solution to σk(g−1 ◦ Ag) = const., normalized to be 2−kn

k

• , the linearization of the fully nonlinear PDE (2) at ξ(t)

is then Lξ(φ) + 21−kk n k

• e−2kξ(t)φ(t, θ) = 0.

If we take the projections of φ(t, ·) into spherical harmonics: φ(t, θ) =

• j

φj(t)Yj(θ), where Yj(θ) are the normalized eigenfunctions of ∆θ on L2(Sn−1). Then φj(t) satisfies the ODE Lj[φj] := φ

′′

j (t) + B(t)

A(t)φ

′

j(t) +

• −λj

C(t) A(t) + ne−2kξ(t) 2A(t)(1 − ξ2

t (t))k−2

• φj(t)

= 0, (25)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

where λj are the eigenvalues of ∆θ on L2(Sn−1) associated with Yj(θ), thus λ0 = 0, λ1 = · · · = λn = n − 1, λj ≥ 2n, for j > n.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

where λj are the eigenvalues of ∆θ on L2(Sn−1) associated with Yj(θ), thus λ0 = 0, λ1 = · · · = λn = n − 1, λj ≥ 2n, for j > n. Similar to properties of the linearized operator to the scalar curvature operator used in [KMPS], we have the following properties for the Lj’s. Proposition 3 For all solutions ξh(t) to (3) with h ≥ 0, k < n and j ≥ 1, the following holds: (i) Lj[φ] = 0 has a pair of linearly independent solution basis

• n R, one of which grows unbounded and the other one

decays exponentially as t → ∞;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(ii) Any solution of Lj[φ] = 0 which is bounded for R+ must decay exponentially;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(ii) Any solution of Lj[φ] = 0 which is bounded for R+ must decay exponentially; (iii) Any solution of Lj[φ] = 0 which is bounded for all of R must be identically 0;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(ii) Any solution of Lj[φ] = 0 which is bounded for R+ must decay exponentially; (iii) Any solution of Lj[φ] = 0 which is bounded for all of R must be identically 0; (iv) Any non-zero solution of Lj[φ] = 0 which is bounded for all

• f R+ must be unbounded on R−.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(ii) Any solution of Lj[φ] = 0 which is bounded for R+ must decay exponentially; (iii) Any solution of Lj[φ] = 0 which is bounded for all of R must be identically 0; (iv) Any non-zero solution of Lj[φ] = 0 which is bounded for all

• f R+ must be unbounded on R−.

While Proposition 3 is sufficient for providing a proof for Theorem 1, Theorem 2 requires some more detailed knowledge about the linearized operator to (2).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(ii) Any solution of Lj[φ] = 0 which is bounded for R+ must decay exponentially; (iii) Any solution of Lj[φ] = 0 which is bounded for all of R must be identically 0; (iv) Any non-zero solution of Lj[φ] = 0 which is bounded for all

• f R+ must be unbounded on R−.

While Proposition 3 is sufficient for providing a proof for Theorem 1, Theorem 2 requires some more detailed knowledge about the linearized operator to (2). More specifically, the decay rates of bounded solutions to Lj[φ] = 0 on R+ need to be faster than e−t when λj ≥ 2n.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k < n and h > 0, Lj is an ordinary differential

• perator with period coefficient, so, by Floquet theory, has a set
• f well defined characteristic roots which give the exponential

decay/grow rates to solutions φ of Lj[φ] = 0 on R.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k < n and h > 0, Lj is an ordinary differential

• perator with period coefficient, so, by Floquet theory, has a set
• f well defined characteristic roots which give the exponential

decay/grow rates to solutions φ of Lj[φ] = 0 on R. In fact, Lj[φ] = 0 has a set of fundamental solutions in the form

• f eρjtp1(t) and e−ρjtp2(t) for some periodic functions p1(t) and

p2(t), when ρj = 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

When h = 0, ξ(t) = ln cosh(t), and (25) takes the form of Lj[φj] = φ

′′

j (t) − (n − 2) tanh(t)φ

′

j(t) +

• −λj + n cosh−2(t)
• φj,

(26) so a similar notion as characteristic roots can be defined, which is also the case when 2k = n.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

When h = 0, ξ(t) = ln cosh(t), and (25) takes the form of Lj[φj] = φ

′′

j (t) − (n − 2) tanh(t)φ

′

j(t) +

• −λj + n cosh−2(t)
• φj,

(26) so a similar notion as characteristic roots can be defined, which is also the case when 2k = n. We have the following Lemma 1 When 2k ≤ n and h ≥ 0, there is a β∗ > √ 2 such that for all λj ≥ 2n, the associated ρj satisfies ρj ≥ β∗.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

With such knowledge, we can now establish Proposition 4 Suppose that φ(t, θ) → 0 as t → ∞, uniformly in θ ∈ Sn−1, and satisfies Lξ(φ)+21−kk n k

• e−2kξ(t)φ(t, θ) = r(t, θ),

for t ≥ t0 and θ ∈ Sn−1. (27)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

With such knowledge, we can now establish Proposition 4 Suppose that φ(t, θ) → 0 as t → ∞, uniformly in θ ∈ Sn−1, and satisfies Lξ(φ)+21−kk n k

• e−2kξ(t)φ(t, θ) = r(t, θ),

for t ≥ t0 and θ ∈ Sn−1. (27) Suppose that for some 0 < β < β∗ and β = 1, |r(t, θ)| e−βt.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

With such knowledge, we can now establish Proposition 4 Suppose that φ(t, θ) → 0 as t → ∞, uniformly in θ ∈ Sn−1, and satisfies Lξ(φ)+21−kk n k

• e−2kξ(t)φ(t, θ) = r(t, θ),

for t ≥ t0 and θ ∈ Sn−1. (27) Suppose that for some 0 < β < β∗ and β = 1, |r(t, θ)| e−βt. Then there exist constants aj for j = 1, · · · , n, such that |φ(t, θ) −

n

• j=1

aje−t(1 + ξt(t))Yj(θ)| e−βt. (28)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In fact, when β∗ ≤ β < ρn+1, (28) continues to hold,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In fact, when β∗ ≤ β < ρn+1, (28) continues to hold, and when β > ρn+1, we will have |φ(t, θ) −

n

• j=1

aje−t(1 + ξt(t))Yj(θ)| e−ρn+1t. (29)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In fact, when β∗ ≤ β < ρn+1, (28) continues to hold, and when β > ρn+1, we will have |φ(t, θ) −

n

• j=1

aje−t(1 + ξt(t))Yj(θ)| e−ρn+1t. (29) When β = 1, (28) continues to hold if the right hand side is modified into te−t;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In fact, when β∗ ≤ β < ρn+1, (28) continues to hold, and when β > ρn+1, we will have |φ(t, θ) −

n

• j=1

aje−t(1 + ξt(t))Yj(θ)| e−ρn+1t. (29) When β = 1, (28) continues to hold if the right hand side is modified into te−t;and when β = ρn+1, (29) continues to hold if the right hand side is modified into te−ρn+1t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We now provide a proof for Theorem 2. Our proof is very much like the one in [KMPS] for the k = 1 case, once we have

• btained the needed linear analysis.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We now provide a proof for Theorem 2. Our proof is very much like the one in [KMPS] for the k = 1 case, once we have

• btained the needed linear analysis.

Proof of Theorem 2. Our starting point is still Lξh(·+τ)(φ) + Q(φ) + 2kce−2kξh(t+τ)φ(t, θ) = 0, where φ(t, θ) = w(t, θ) − ξh(t + τ), and our premise is: |Q(φ)| e−2αt whenever we have |φ, ∂φ, ∂2φ| e−αt. (30)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We now provide a proof for Theorem 2. Our proof is very much like the one in [KMPS] for the k = 1 case, once we have

• btained the needed linear analysis.

Proof of Theorem 2. Our starting point is still Lξh(·+τ)(φ) + Q(φ) + 2kce−2kξh(t+τ)φ(t, θ) = 0, where φ(t, θ) = w(t, θ) − ξh(t + τ), and our premise is: |Q(φ)| e−2αt whenever we have |φ, ∂φ, ∂2φ| e−αt. (30) In Theorem 1 we already established Step 1. For some α0 > 0, |φ, ∂φ, ∂2φ| e−α0t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

If α0 ≥ ρn+1, we stop and have now proved |w(t, θ) − ξh(t + τ)| = |φ(t, θ)| e−ρn+1t, where ρn+1 > √ 2; if 1 < α0 < ρn+1, we jump to Step 3; if α0 ≤ 1, we move onto

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

If α0 ≥ ρn+1, we stop and have now proved |w(t, θ) − ξh(t + τ)| = |φ(t, θ)| e−ρn+1t, where ρn+1 > √ 2; if 1 < α0 < ρn+1, we jump to Step 3; if α0 ≤ 1, we move onto Step 2. Recall that we now have |Q(φ)| e−2α0t. If 2α0 > ρn+1, then we can apply Proposition 4 directly to conclude our proof;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

If α0 ≥ ρn+1, we stop and have now proved |w(t, θ) − ξh(t + τ)| = |φ(t, θ)| e−ρn+1t, where ρn+1 > √ 2; if 1 < α0 < ρn+1, we jump to Step 3; if α0 ≤ 1, we move onto Step 2. Recall that we now have |Q(φ)| e−2α0t. If 2α0 > ρn+1, then we can apply Proposition 4 directly to conclude our proof; If 1 < 2α0 ≤ ρn+1,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

If α0 ≥ ρn+1, we stop and have now proved |w(t, θ) − ξh(t + τ)| = |φ(t, θ)| e−ρn+1t, where ρn+1 > √ 2; if 1 < α0 < ρn+1, we jump to Step 3; if α0 ≤ 1, we move onto Step 2. Recall that we now have |Q(φ)| e−2α0t. If 2α0 > ρn+1, then we can apply Proposition 4 directly to conclude our proof; If 1 < 2α0 ≤ ρn+1, then we certainly still have |Q(φ)| e−2αt for some 1 < 2α < ρn+1 and can apply Proposition 4 to imply that |w(t, θ) − ξh(t + τ) −

n

• j=1

aje−(t+τ)(1 + ξ

′

h(t + τ))Yj(θ)| e−2αt,

(31) for some constants aj for j = 1, · · · , n, and jump to Step 3;

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

if 2α0 ≤ 1, we may take α0 to satisfy 2α0 < 1 and apply Proposition 4 to imply that |φ(t, θ) −

n

• j=1

aje−(t+τ)(1 + ξ

′

h(t + τ))Yj(θ)| e−2α0t

for some constants aj for j = 1, · · · , n.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

if 2α0 ≤ 1, we may take α0 to satisfy 2α0 < 1 and apply Proposition 4 to imply that |φ(t, θ) −

n

• j=1

aje−(t+τ)(1 + ξ

′

h(t + τ))Yj(θ)| e−2α0t

for some constants aj for j = 1, · · · , n. This certainly implies that |φ(t, θ)| e−2α0t. (32)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

if 2α0 ≤ 1, we may take α0 to satisfy 2α0 < 1 and apply Proposition 4 to imply that |φ(t, θ) −

n

• j=1

aje−(t+τ)(1 + ξ

′

h(t + τ))Yj(θ)| e−2α0t

for some constants aj for j = 1, · · · , n. This certainly implies that |φ(t, θ)| e−2α0t. (32) Next we use higher derivative estimates for w(t, θ) and ξh(t + τ) and interpolation with (32) to obtain |φ, ∂φ, ∂2φ| e−2α′t for any α′ < α0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Now we go back to the beginning of step 2 and repeat the process with a new α1 > α0 to replace the α0 there, say, α1 = 1.8α0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Now we go back to the beginning of step 2 and repeat the process with a new α1 > α0 to replace the α0 there, say, α1 = 1.8α0. After a finite number of steps, we will reach a stage where 2α > 1 and ready to move onto

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Now we go back to the beginning of step 2 and repeat the process with a new α1 > α0 to replace the α0 there, say, α1 = 1.8α0. After a finite number of steps, we will reach a stage where 2α > 1 and ready to move onto Step 3. At this stage, we have |φ(t, θ)| e−t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Now we go back to the beginning of step 2 and repeat the process with a new α1 > α0 to replace the α0 there, say, α1 = 1.8α0. After a finite number of steps, we will reach a stage where 2α > 1 and ready to move onto Step 3. At this stage, we have |φ(t, θ)| e−t. Repeating the last part of Step 2 involving the derivative estimates for w(t, θ) and ξh(t + τ) to bootstrap the estimate for Q(φ) to |Q(φ)| e−αt, with α can be as close to 2 as one needs.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Now we go back to the beginning of step 2 and repeat the process with a new α1 > α0 to replace the α0 there, say, α1 = 1.8α0. After a finite number of steps, we will reach a stage where 2α > 1 and ready to move onto Step 3. At this stage, we have |φ(t, θ)| e−t. Repeating the last part of Step 2 involving the derivative estimates for w(t, θ) and ξh(t + τ) to bootstrap the estimate for Q(φ) to |Q(φ)| e−αt, with α can be as close to 2 as one needs. Then, depending on whether ρn+1 ≥ 2 or otherwise, one can apply Proposition 4 to obtain (28) or (29).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the first case, we can continue the iteration until 2α > 2.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the first case, we can continue the iteration until 2α > 2. But due to the presence of e−(t+τ)(1 + ξ

′

h(t + τ))Yj(θ) in the

estimate for φ, the estimate for Q(φ) can not be better than e−2t. This explains the appearance of min{2, ρn+1} in (14).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Using (18) and (19), we can obtain, in addition to (21), e(2k−n)γ (1 − γ2

t )k + η3(t)

• − e−nγ {1 + η4(t)} = h,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Using (18) and (19), we can obtain, in addition to (21), e(2k−n)γ (1 − γ2

t )k + η3(t)

• − e−nγ {1 + η4(t)} = h,

also 2(1 − γ2

t )k−1

k nγtt + n − 2k 2n (1 − γ2

t )

• + η1(t)

=e−2kγ (1 + η2(t)) , (33) where ηi(t), for i = 1, 2, have the decay rate ηi(t) = O(e−2(1−δ)t) as t → ∞, where δ > 0 can be made as small as one needs, as in (19).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Using (18) and (19), we can obtain, in addition to (21), e(2k−n)γ (1 − γ2

t )k + η3(t)

• − e−nγ {1 + η4(t)} = h,

also 2(1 − γ2

t )k−1

k nγtt + n − 2k 2n (1 − γ2

t )

• + η1(t)

=e−2kγ (1 + η2(t)) , (33) where ηi(t), for i = 1, 2, have the decay rate ηi(t) = O(e−2(1−δ)t) as t → ∞, where δ > 0 can be made as small as one needs, as in (19). We will use (18), (21),and (33) to deduce that γ(t), therefore w(t, θ), has the desired asymptotic behavior.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We first handle the case h = 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We first handle the case h = 0. It follows from (21) and h = 0 that, for sufficiently large t, γt(t) = 0 can occur only near γ(t) = 0,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We first handle the case h = 0. It follows from (21) and h = 0 that, for sufficiently large t, γt(t) = 0 can occur only near γ(t) = 0, which, together with (ii) above, implies that γt(t) > 0 for suffciently large t and limt→∞ γ(t) = ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

We first handle the case h = 0. It follows from (21) and h = 0 that, for sufficiently large t, γt(t) = 0 can occur only near γ(t) = 0, which, together with (ii) above, implies that γt(t) > 0 for suffciently large t and limt→∞ γ(t) = ∞. Putting this information back into (21), we see that 1 − γ2

t (t) =: η(t) → 0 as

t → ∞. Since γt(t) > 0 for sufficiently large t, we conclude that 1 − γt(t) → 0 as t → ∞. As a consequence, γ(t) ≥ (1 − ǫ)t + γ0 for large t and some ǫ > 0 small and γ0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

This would imply through (21) that |η(t)| ≤ Ce− 2(1−δ)

k

t

for some constant C > 0 and for large t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

This would imply through (21) that |η(t)| ≤ Ce− 2(1−δ)

k

t

for some constant C > 0 and for large t. Finally, we have |γt(t) − 1| = |

• 1 − η(t) − 1| ≤ Ce− 2(1−δ)

k

t,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

This would imply through (21) that |η(t)| ≤ Ce− 2(1−δ)

k

t

for some constant C > 0 and for large t. Finally, we have |γt(t) − 1| = |

• 1 − η(t) − 1| ≤ Ce− 2(1−δ)

k

t,

from which we conclude that γ(t) − t = c + O(e− 2(1−δ)

k

t),

for some c as t → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

This would imply through (21) that |η(t)| ≤ Ce− 2(1−δ)

k

t

for some constant C > 0 and for large t. Finally, we have |γt(t) − 1| = |

• 1 − η(t) − 1| ≤ Ce− 2(1−δ)

k

t,

from which we conclude that γ(t) − t = c + O(e− 2(1−δ)

k

t),

for some c as t → ∞. Note that ξ0(t), the solution to (5), to which (33) is a perturbation, with h = 0, satisfies ξ0(t) = t − ln 2 + O(e−2t).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

This would imply through (21) that |η(t)| ≤ Ce− 2(1−δ)

k

t

for some constant C > 0 and for large t. Finally, we have |γt(t) − 1| = |

• 1 − η(t) − 1| ≤ Ce− 2(1−δ)

k

t,

from which we conclude that γ(t) − t = c + O(e− 2(1−δ)

k

t),

for some c as t → ∞. Note that ξ0(t), the solution to (5), to which (33) is a perturbation, with h = 0, satisfies ξ0(t) = t − ln 2 + O(e−2t). Therefore w(t, θ) − t = γ(t) + w(t, θ) = ξ0(t + c + ln 2) + O(e−(1−δ)t), as t → ∞,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

and u(x) = e− n−2

2 (w(t,θ)−t) = e− n−2 2 (γ(t)−t+

w(t,θ)).

As a consequence

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

and u(x) = e− n−2

2 (w(t,θ)−t) = e− n−2 2 (γ(t)−t+

w(t,θ)).

As a consequence lim

x→0 u(x) = e− n−2

2 c =: u(0) > 0

exists, with |u(x) − u(0)| ≤ |u(0)|

• e

− n−2

2

• w(t,θ)+O(e− 2(1−δ)

k t)

• − 1
• ≤ Ce− 2(1−δ)

k

t

≤ C|x|

2(1−δ) k

.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

and u(x) = e− n−2

2 (w(t,θ)−t) = e− n−2 2 (γ(t)−t+

w(t,θ)).

As a consequence lim

x→0 u(x) = e− n−2

2 c =: u(0) > 0

exists, with |u(x) − u(0)| ≤ |u(0)|

• e

− n−2

2

• w(t,θ)+O(e− 2(1−δ)

k t)

• − 1
• ≤ Ce− 2(1−δ)

k

t

≤ C|x|

2(1−δ) k

. The case h > 0 and 2k > n is argued in a similar way. We next describe how to handle the two cases: 2k < n and h > 0,

• r 2k = n and h > 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k = n and h > 0, we use the property 1 − γ2

t (t) + \

• Sn−1 |∇

w(t, θ)|2dθ ≥ 0, and (21) to prove that 0 ≤ h ≤ 1. The possibility of h = 1 can be ruled out after a more careful examination.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k = n and h > 0, we use the property 1 − γ2

t (t) + \

• Sn−1 |∇

w(t, θ)|2dθ ≥ 0, and (21) to prove that 0 ≤ h ≤ 1. The possibility of h = 1 can be ruled out after a more careful examination. Then we argue that γt(t) > 0 for t sufficiently large.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k = n and h > 0, we use the property 1 − γ2

t (t) + \

• Sn−1 |∇

w(t, θ)|2dθ ≥ 0, and (21) to prove that 0 ≤ h ≤ 1. The possibility of h = 1 can be ruled out after a more careful examination. Then we argue that γt(t) > 0 for t sufficiently large. Next (21) implies that e−nγ(t) = O(e−αt) as t → ∞ for some α > 0 depending on 0 < h < 1.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k = n and h > 0, we use the property 1 − γ2

t (t) + \

• Sn−1 |∇

w(t, θ)|2dθ ≥ 0, and (21) to prove that 0 ≤ h ≤ 1. The possibility of h = 1 can be ruled out after a more careful examination. Then we argue that γt(t) > 0 for t sufficiently large. Next (21) implies that e−nγ(t) = O(e−αt) as t → ∞ for some α > 0 depending on 0 < h < 1. Going back to (21), we find ηk(t) = h + O(e−αt) as t → ∞, and γt(t) =

• 1 − η(t) =
• 1 −

k

√ h + O(e−αt),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

which implies that γ(t) =

• 1 −

k

√ ht + γ0 + O(e−αt), for some γ0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

which implies that γ(t) =

• 1 −

k

√ ht + γ0 + O(e−αt), for some γ0. Note that in the case 2k = n, ξh(t), solution to (5), the unperturbed (33), satisfies ξh(t) =

• 1 −

k

√ ht + c + O(e−αt) for some c and α > 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

which implies that γ(t) =

• 1 −

k

√ ht + γ0 + O(e−αt), for some γ0. Note that in the case 2k = n, ξh(t), solution to (5), the unperturbed (33), satisfies ξh(t) =

• 1 −

k

√ ht + c + O(e−αt) for some c and α > 0. So we can choose τ such that |γ(t) − ξh(t + τ)| = O(e−αt), and |w(t, θ) − ξh(t + τ)| = O(e−αt).

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Now with u(x) = e− n−2

2 (w(t,θ)−t) = e− n−2 2 (γ(t)−t+

w(t,θ)),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Now with u(x) = e− n−2

2 (w(t,θ)−t) = e− n−2 2 (γ(t)−t+

w(t,θ)),

we find |x|

n−2 2

• 1−

√

1− k √ h

• u(x) = e

− n−2

2

• γ(t)−

√

1− k √ ht+ w(t,θ)

• ,

which extends to a Cα(BR) positive function for some α > 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k < n and h > 0, we will prove that there is a solution ξh(t) to (5) some α > 0 and τ such that γ(t) − ξh(t + τ) = O(e−αt) as t → ∞,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k < n and h > 0, we will prove that there is a solution ξh(t) to (5) some α > 0 and τ such that γ(t) − ξh(t + τ) = O(e−αt) as t → ∞, thus |w(t, θ) − ξh(t + τ)| = O(e−αt) as t → ∞,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k < n and h > 0, we will prove that there is a solution ξh(t) to (5) some α > 0 and τ such that γ(t) − ξh(t + τ) = O(e−αt) as t → ∞, thus |w(t, θ) − ξh(t + τ)| = O(e−αt) as t → ∞,and u(x) = |x|− n−2

2 e− n−2 2 ξh(t+τ) (1 + O(|x|α))

as x → 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

In the case 2k < n and h > 0, we will prove that there is a solution ξh(t) to (5) some α > 0 and τ such that γ(t) − ξh(t + τ) = O(e−αt) as t → ∞, thus |w(t, θ) − ξh(t + τ)| = O(e−αt) as t → ∞,and u(x) = |x|− n−2

2 e− n−2 2 ξh(t+τ) (1 + O(|x|α))

as x → 0. This would be similar to the approach in [CGS], where they state that it follows from (7.14) β

′2 =

n − 2 n 2 β2 − n − 2 n β

2n n−2 + D∞ +

• β2 + β

′2

O(e−t), (7.14) that, when D∞ = D < 0,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

β is asymptotic to a suitable translate of the solution ψ of (1.6) ψ

′2 =

n − 2 n 2 ψ2 − n − 2 n ψ

2n n−2 + D.

(1.6)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

β is asymptotic to a suitable translate of the solution ψ of (1.6) ψ

′2 =

n − 2 n 2 ψ2 − n − 2 n ψ

2n n−2 + D.

(1.6) We will provide a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and σk curvature cases.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Consider a solution β(t) to β

′′(t) = f(β′(t), β(t)) + e1(t),

t ≥ 0, (34) where f is locally Lipschitz, and e1(t) is considered as a perturbation term with e1(t) → 0 as t → ∞ at a sufficiently fast rate to be specified later.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Consider a solution β(t) to β

′′(t) = f(β′(t), β(t)) + e1(t),

t ≥ 0, (34) where f is locally Lipschitz, and e1(t) is considered as a perturbation term with e1(t) → 0 as t → ∞ at a sufficiently fast rate to be specified later. Suppose that |β(t)| + |β′(t)| is bounded over t ∈ [0, ∞). Then by a compactness argument there exists a solution ψ(t) to ψ

′′(t) = f(ψ′(t), ψ(t))

(35) which exists for all t ∈ R

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

such that for a sequence of ti → ∞, for any finite L, β(ti + τ) − ψ(τ) → 0, in C1[−L, L], (36) as i → ∞.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

such that for a sequence of ti → ∞, for any finite L, β(ti + τ) − ψ(τ) → 0, in C1[−L, L], (36) as i → ∞. ODE Stability Theorem Suppose that, for the β(t) and ψ(t) above, ψ(t) is a periodic solution to (35) with (minimal) period T > 0. Thus for some finite m < M, ψ(R) = [m, M]. We may do a time translation for ψ(t) so that ψ(0) = m, ψ′(0) = 0, then the approximation property (36) can be reformulated as: there exists some s such that for any finite L, as i → ∞, β(ti + τ) − ψ(−s + τ) → 0, in C1[0, L]. (37)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Suppose that ψ(t) has a first integral in the form of H(ψ′(t), ψ(t)) = 0, (38) for some continuous function H(x, y),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Suppose that ψ(t) has a first integral in the form of H(ψ′(t), ψ(t)) = 0, (38) for some continuous function H(x, y), where H satisfies the following non-degeneracy condition: for some ǫ1 > 0, A > 0 and l > 0, |H(0, y)| = |H(0, y) − H(0, m)| ≥ A|y − m|l, (39) for any y with |y − m| ≤ ǫ1.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Suppose that ψ(t) has a first integral in the form of H(ψ′(t), ψ(t)) = 0, (38) for some continuous function H(x, y), where H satisfies the following non-degeneracy condition: for some ǫ1 > 0, A > 0 and l > 0, |H(0, y)| = |H(0, y) − H(0, m)| ≥ A|y − m|l, (39) for any y with |y − m| ≤ ǫ1. Suppose also that β(t) has H as an approximate first integral |H(β

′(t), β(t))| ≤ e2(t),

for t ≥ 0, (40)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

where e2(t) → 0 as t → ∞. Without loss of generality, we may suppose that e2(t) is monotone non-increasing in t.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

where e2(t) → 0 as t → ∞. Without loss of generality, we may suppose that e2(t) is monotone non-increasing in t. Finally suppose that ∞

• (e2(t))1/k + sup

τ≥t

|e1(τ)|

• dt < ∞.

(41)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

where e2(t) → 0 as t → ∞. Without loss of generality, we may suppose that e2(t) is monotone non-increasing in t. Finally suppose that ∞

• (e2(t))1/k + sup

τ≥t

|e1(τ)|

• dt < ∞.

(41) Then, for some s∞, |β(t) − ψ(t − s∞)| + |β

′(t) − ψ ′(t − s∞)| → 0,

as t → ∞. (42)

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Remarks Our proof in fact gives a decay rate for |β(t) − ψ(t − s∞)| + |β

′ − ψ ′(t − s∞)| in terms of

∞

t−1

• e2(τ)1/k + maxτ ′≥τ |e1(τ ′)|
• dτ, which is exponentially

decaying when |e1(t)| and e2(t) have exponential decay rate.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Remarks Our proof in fact gives a decay rate for |β(t) − ψ(t − s∞)| + |β

′ − ψ ′(t − s∞)| in terms of

∞

t−1

• e2(τ)1/k + maxτ ′≥τ |e1(τ ′)|
• dτ, which is exponentially

decaying when |e1(t)| and e2(t) have exponential decay rate. The case that ψ(t) is a constant solution (i.e. period 0 solution) can be incorporated into the theorem easily.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

Remarks Our proof in fact gives a decay rate for |β(t) − ψ(t − s∞)| + |β

′ − ψ ′(t − s∞)| in terms of

∞

t−1

• e2(τ)1/k + maxτ ′≥τ |e1(τ ′)|
• dτ, which is exponentially

decaying when |e1(t)| and e2(t) have exponential decay rate. The case that ψ(t) is a constant solution (i.e. period 0 solution) can be incorporated into the theorem easily. To complete our proof in the case 2k < n and h > 0, we note that the upper bound (15) in Proposition 2 and the approximate first integral (21) imply that γ ≤ γ(t) ≤ γ for some finite γ ≤ γ.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(21) further implies that 1 − γ2

t (t) ≥ ǫ

for some ǫ > 0.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(21) further implies that 1 − γ2

t (t) ≥ ǫ

for some ǫ > 0. If we take f(γ, γt) = ne−2kγ 2k(1 − γ2

t )k−1 − n − 2k

2k (1 − γ2

t ),

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(21) further implies that 1 − γ2

t (t) ≥ ǫ

for some ǫ > 0. If we take f(γ, γt) = ne−2kγ 2k(1 − γ2

t )k−1 − n − 2k

2k (1 − γ2

t ),

and H(γ, γt) = e(2k−n)γ(1 − γ2

t )k − e−nγ − h,

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated

Description of the problem Statements of Theorems Discussion of proof Outline of key steps in proof More details on the spectrum analysis Spectrum analysis for higher order expansion Proof for Theorem 2 Another proof using approximating ODE An asymptotic result on ODE

(21) further implies that 1 − γ2

t (t) ≥ ǫ

for some ǫ > 0. If we take f(γ, γt) = ne−2kγ 2k(1 − γ2

t )k−1 − n − 2k

2k (1 − γ2

t ),

and H(γ, γt) = e(2k−n)γ(1 − γ2

t )k − e−nγ − h,

then the conditions in the ODE Stability Theorems are satisfied. Therefore, we can conclude the proof of our Main Theorem.

Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σk -Yamabe equation near isolated