Regularized determinants and conformally invariant operators M. - - PowerPoint PPT Presentation

Regularized determinants and conformally invariant

• perators
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA)

Workshop on Geometric Partial Differential Equations

June 7, 2012

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 1 / 30

Regularized determinants

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 2 / 30

Regularized determinants

(Mn, g) a closed manifold.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 2 / 30

Regularized determinants

(Mn, g) a closed manifold. Let 0 = λ0 < λ1 ≤ λ2 ≤ · · · denote the eigenvalues of (−∆g), counting multiplicities.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 2 / 30

Regularized determinants

(Mn, g) a closed manifold. Let 0 = λ0 < λ1 ≤ λ2 ≤ · · · denote the eigenvalues of (−∆g), counting multiplicities. We want to define: det(−∆g).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 2 / 30

Regularized determinants

(Mn, g) a closed manifold. Let 0 = λ0 < λ1 ≤ λ2 ≤ · · · denote the eigenvalues of (−∆g), counting multiplicities. We want to define: det(−∆g).

1 By Weyl’s law, λj ∼ j2/n.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 2 / 30

Regularized determinants

(Mn, g) a closed manifold. Let 0 = λ0 < λ1 ≤ λ2 ≤ · · · denote the eigenvalues of (−∆g), counting multiplicities. We want to define: det(−∆g).

1 By Weyl’s law, λj ∼ j2/n. 2 Define the spectral zeta-function

ζ(s) =

• j≥1

λ−s

j ,

Re(s) > n/2.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 2 / 30

det(−∆)

Formally, if we differentiate

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 3 / 30

det(−∆)

Formally, if we differentiate ζ(s) =

• j≥1

λ−s

j

⇒

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 3 / 30

det(−∆)

Formally, if we differentiate ζ(s) =

• j≥1

λ−s

j

⇒ ζ′(0) = −

• j≥1

(log λj) = − log

j≥1

λj

• ,
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 3 / 30

det(−∆)

Formally, if we differentiate ζ(s) =

• j≥1

λ−s

j

⇒ ζ′(0) = −

• j≥1

(log λj) = − log

j≥1

λj

• ,

hence ”Definition.” det(−∆g) = e−ζ′(0).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 3 / 30

det(−∆)

Formally, if we differentiate ζ(s) =

• j≥1

λ−s

j

⇒ ζ′(0) = −

• j≥1

(log λj) = − log

j≥1

λj

• ,

hence ”Definition.” det(−∆g) = e−ζ′(0).

• Using heat kernel asymptotics, ζ can be meromorphically continued to

be regular at s = 0. In particular, ζ′(0) is well defined.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 3 / 30

Example: (Σ, g) a surface

Using the Gamma Function, write λ−s

j

= 1 Γ(s) ∞ e−λjtts−1 dt,

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 4 / 30

Example: (Σ, g) a surface

Using the Gamma Function, write λ−s

j

= 1 Γ(s) ∞ e−λjtts−1 dt, hence ζ(s) = 1 Γ(s) ∞

j≥1

e−λjt ts−1 dt

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 4 / 30

Example: (Σ, g) a surface

Using the Gamma Function, write λ−s

j

= 1 Γ(s) ∞ e−λjtts−1 dt, hence ζ(s) = 1 Γ(s) ∞

j≥1

e−λjt ts−1 dt = 1 Γ(s) ∞

• trL2et∆ − 1
• ts−1 dt.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 4 / 30

Example: (Σ, g) a surface

Using the Gamma Function, write λ−s

j

= 1 Γ(s) ∞ e−λjtts−1 dt, hence ζ(s) = 1 Γ(s) ∞

j≥1

e−λjt ts−1 dt = 1 Γ(s) ∞

• trL2et∆ − 1
• ts−1 dt.

Using well known heat kernel asymptotics = 1 Γ(s) ∞ 1 4πt + K(x) 12π + O(t)

• ts−1 dt
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 4 / 30

Example: (Σ, g) a surface

Using the Gamma Function, write λ−s

j

= 1 Γ(s) ∞ e−λjtts−1 dt, hence ζ(s) = 1 Γ(s) ∞

j≥1

e−λjt ts−1 dt = 1 Γ(s) ∞

• trL2et∆ − 1
• ts−1 dt.

Using well known heat kernel asymptotics = 1 Γ(s) ∞ 1 4πt + K(x) 12π + O(t)

• ts−1 dt

ζ(s) = 1 Γ(s) Area(Σ) 4π(s − 1) + χ(Σ) 6 − 1

• + (holom. in s)
• ,

so ζ′(0) is well defined.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 4 / 30

Polyakov’s Formula

This gives a closed expression for the ratio of the determinants for two conformal metrics on a closed surface.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 5 / 30

Polyakov’s Formula

This gives a closed expression for the ratio of the determinants for two conformal metrics on a closed surface. Theorem (Polyakov, ’81) If ˆ g = e2wg, then log det(−∆ˆ

g)

det(−∆g) = − 1 12π

• M
• |∇w|2 + 2Kw
• dA,

where K = Gauss curvature of g.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 5 / 30

Polyakov’s Formula

This gives a closed expression for the ratio of the determinants for two conformal metrics on a closed surface. Theorem (Polyakov, ’81) If ˆ g = e2wg, then log det(−∆ˆ

g)

det(−∆g) = − 1 12π

• M
• |∇w|2 + 2Kw
• dA,

where K = Gauss curvature of g.

• This forumla naturally defines an action on the space of unit-volume

conformal metrics.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 5 / 30

The work of Osgood-Phillips-Sarnak

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 6 / 30

The work of Osgood-Phillips-Sarnak

Consider the following scale-invariant version of log det: S[w] = − 1 12π

M

• |∇w|2 + 2Kw
• dA −

K dA

• log

M

e2w dA

• .
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 6 / 30

The work of Osgood-Phillips-Sarnak

Consider the following scale-invariant version of log det: S[w] = − 1 12π

M

• |∇w|2 + 2Kw
• dA −

K dA

• log

M

e2w dA

• .

1 If Area(e2wg) = Area(g), then S[w] = log

det(−∆e2w g) det(−∆g) .

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 6 / 30

The work of Osgood-Phillips-Sarnak

Consider the following scale-invariant version of log det: S[w] = − 1 12π

M

• |∇w|2 + 2Kw
• dA −

K dA

• log

M

e2w dA

• .

1 If Area(e2wg) = Area(g), then S[w] = log

det(−∆e2w g) det(−∆g) .

2 S[w + c] = S[w].

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 6 / 30

The work of Osgood-Phillips-Sarnak

Consider the following scale-invariant version of log det: S[w] = − 1 12π

M

• |∇w|2 + 2Kw
• dA −

K dA

• log

M

e2w dA

• .

1 If Area(e2wg) = Area(g), then S[w] = log

det(−∆e2w g) det(−∆g) .

2 S[w + c] = S[w]. 3 Trudinger inequality gives W 1,2 ֒

→ eL2, hence S : W 1,2 → R.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 6 / 30

The work of Osgood-Phillips-Sarnak

Consider the following scale-invariant version of log det: S[w] = − 1 12π

M

• |∇w|2 + 2Kw
• dA −

K dA

• log

M

e2w dA

• .

1 If Area(e2wg) = Area(g), then S[w] = log

det(−∆e2w g) det(−∆g) .

2 S[w + c] = S[w]. 3 Trudinger inequality gives W 1,2 ֒

→ eL2, hence S : W 1,2 → R. Theorem (O-P-S, ’88) S is maximized by a metric of constant (Gauss) curvature in each conformal class.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 6 / 30

The case of (S2, g0)

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 7 / 30

The case of (S2, g0)

For S2 with the round metric we have S[w] = − 1 12π

S2

• |∇w|2 + 2w
• dA − 4π log

S2 e2w dA

• .
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 7 / 30

The case of (S2, g0)

For S2 with the round metric we have S[w] = − 1 12π

S2

• |∇w|2 + 2w
• dA − 4π log

S2 e2w dA

• .

1 The analysis of this case actually precedes the work of OPS, and is

due to Onofri.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 7 / 30

The case of (S2, g0)

For S2 with the round metric we have S[w] = − 1 12π

S2

• |∇w|2 + 2w
• dA − 4π log

S2 e2w dA

• .

1 The analysis of this case actually precedes the work of OPS, and is

due to Onofri.

2 Critical points of S correspond to constant curvature metrics; hence

are given by the orbit of g0 under the action of the conformal group.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 7 / 30

The case of (S2, g0)

For S2 with the round metric we have S[w] = − 1 12π

S2

• |∇w|2 + 2w
• dA − 4π log

S2 e2w dA

• .

1 The analysis of this case actually precedes the work of OPS, and is

due to Onofri.

2 Critical points of S correspond to constant curvature metrics; hence

are given by the orbit of g0 under the action of the conformal group.

3 By Trudinger, Moser inequalities ⇒ S[w] ≤ C.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 7 / 30

The case of (S2, g0)

For S2 with the round metric we have S[w] = − 1 12π

S2

• |∇w|2 + 2w
• dA − 4π log

S2 e2w dA

• .

1 The analysis of this case actually precedes the work of OPS, and is

due to Onofri.

2 Critical points of S correspond to constant curvature metrics; hence

are given by the orbit of g0 under the action of the conformal group.

3 By Trudinger, Moser inequalities ⇒ S[w] ≤ C.

Theorem (Onofri, ’82) S[w] ≤ 0, with equality if and only if e2wg0 = ϕ∗g0.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 7 / 30

Some additional remarks about Polakov’s formula

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 8 / 30

Some additional remarks about Polakov’s formula

1 As pointed out by B. Chow (JDG, ’91), the gradient flow for the

functional S is (up to a sign) the Ricci flow.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 8 / 30

Some additional remarks about Polakov’s formula

1 As pointed out by B. Chow (JDG, ’91), the gradient flow for the

functional S is (up to a sign) the Ricci flow.

2 The key property exploited by Polyakov to derive his formula: In

dimension two, ∆e2wg = e−2w∆g. This does not hold in dimension n ≥ 3.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 8 / 30

Some additional remarks about Polakov’s formula

1 As pointed out by B. Chow (JDG, ’91), the gradient flow for the

functional S is (up to a sign) the Ricci flow.

2 The key property exploited by Polyakov to derive his formula: In

dimension two, ∆e2wg = e−2w∆g. This does not hold in dimension n ≥ 3.

3 Let

Lg = ∆g − (n − 2) 4(n − 1)Rg, where R = Rg is the scalar curvature of g. This is called the conformal Laplacian. Note

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 8 / 30

Some additional remarks about Polakov’s formula

1 As pointed out by B. Chow (JDG, ’91), the gradient flow for the

functional S is (up to a sign) the Ricci flow.

2 The key property exploited by Polyakov to derive his formula: In

dimension two, ∆e2wg = e−2w∆g. This does not hold in dimension n ≥ 3.

3 Let

Lg = ∆g − (n − 2) 4(n − 1)Rg, where R = Rg is the scalar curvature of g. This is called the conformal Laplacian. Note ˆ g = e2wg ⇒ Lˆ

gφ = e− n+2

2 wLg(e n−2 2 wφ).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 8 / 30

Some additional remarks about Polakov’s formula

1 As pointed out by B. Chow (JDG, ’91), the gradient flow for the

functional S is (up to a sign) the Ricci flow.

2 The key property exploited by Polyakov to derive his formula: In

dimension two, ∆e2wg = e−2w∆g. This does not hold in dimension n ≥ 3.

3 Let

Lg = ∆g − (n − 2) 4(n − 1)Rg, where R = Rg is the scalar curvature of g. This is called the conformal Laplacian. Note ˆ g = e2wg ⇒ Lˆ

gφ = e− n+2

2 wLg(e n−2 2 wφ). 4 In higher dimensions there are many other examples of conformally

invariant operators...

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 8 / 30

Four dimensions: the work of Branson-Ørsted

(M, g) a closed four-manifold.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 9 / 30

Four dimensions: the work of Branson-Ørsted

(M, g) a closed four-manifold. Suppose A = Ag is conformally invariant:

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 9 / 30

Four dimensions: the work of Branson-Ørsted

(M, g) a closed four-manifold. Suppose A = Ag is conformally invariant: Ae2wgφ = eabAg(ebwφ) (and satisfies some additional technical assumptions).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 9 / 30

Four dimensions: the work of Branson-Ørsted

(M, g) a closed four-manifold. Suppose A = Ag is conformally invariant: Ae2wgφ = eabAg(ebwφ) (and satisfies some additional technical assumptions). For several important examples, Branson-Ørsted gave a Polyakov-type formula for FA[w] = log det Aˆ

g

det Ag , ˆ g = e2wg.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 9 / 30

Four dimensions: the work of Branson-Ørsted

(M, g) a closed four-manifold. Suppose A = Ag is conformally invariant: Ae2wgφ = eabAg(ebwφ) (and satisfies some additional technical assumptions). For several important examples, Branson-Ørsted gave a Polyakov-type formula for FA[w] = log det Aˆ

g

det Ag , ˆ g = e2wg. Note: In the following, we assume A has a trivial kernel; otherwise the forgoing formulas need to be modified.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 9 / 30

The formula of Branson-Ørsted, cont.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 10 / 30

The formula of Branson-Ørsted, cont.

• First observation: FA is always a linear combination of the same three

terms:

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 10 / 30

The formula of Branson-Ørsted, cont.

• First observation: FA is always a linear combination of the same three

terms: i.e., FA[w] = γ1(A) · I[w] + γ2(A) · II[w] + γ3(A) · III[w], where I, II, III are functionals on the space of conformal metrics.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 10 / 30

The formula of Branson-Ørsted, cont.

• First observation: FA is always a linear combination of the same three

terms: i.e., FA[w] = γ1(A) · I[w] + γ2(A) · II[w] + γ3(A) · III[w], where I, II, III are functionals on the space of conformal metrics.

• Consequently, there are two parts to Branson-Ørsted’s work:
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 10 / 30

The formula of Branson-Ørsted, cont.

• First observation: FA is always a linear combination of the same three

terms: i.e., FA[w] = γ1(A) · I[w] + γ2(A) · II[w] + γ3(A) · III[w], where I, II, III are functionals on the space of conformal metrics.

• Consequently, there are two parts to Branson-Ørsted’s work:

1 Identifying the functionals I, II, III.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 10 / 30

The formula of Branson-Ørsted, cont.

• First observation: FA is always a linear combination of the same three

terms: i.e., FA[w] = γ1(A) · I[w] + γ2(A) · II[w] + γ3(A) · III[w], where I, II, III are functionals on the space of conformal metrics.

• Consequently, there are two parts to Branson-Ørsted’s work:

1 Identifying the functionals I, II, III. 2 For a particular choice of the operator A, computing the coefficients

γ1, γ2, γ3.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 10 / 30

The Branson-Ørsted formula for space forms

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 11 / 30

The Branson-Ørsted formula for space forms

If (M4, g) is flat, then I[w] = 0,

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 11 / 30

The Branson-Ørsted formula for space forms

If (M4, g) is flat, then I[w] = 0, II[w] =

• (∆w)2,
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 11 / 30

The Branson-Ørsted formula for space forms

If (M4, g) is flat, then I[w] = 0, II[w] =

• (∆w)2,

III[w] = 12

• (∆w + |∇w|2)2.

Note that all three are non-negative.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 11 / 30

The Branson-Ørsted formula for space forms

If (M4, g) is flat, then I[w] = 0, II[w] =

• (∆w)2,

III[w] = 12

• (∆w + |∇w|2)2.

Note that all three are non-negative.

• Hence log det formula is a linear combination of three basic functionals:

B[w] =

• (∆w)2, H[w] =
• (∆w)|∇w|2, P[w] =
• |∇w|4,

with gradients

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 11 / 30

The Branson-Ørsted formula for space forms

If (M4, g) is flat, then I[w] = 0, II[w] =

• (∆w)2,

III[w] = 12

• (∆w + |∇w|2)2.

Note that all three are non-negative.

• Hence log det formula is a linear combination of three basic functionals:

B[w] =

• (∆w)2, H[w] =
• (∆w)|∇w|2, P[w] =
• |∇w|4,

with gradients ∇B = ∆2w, ∇H = σ2(∇2w), ∇P = div(|∇w|2∇w).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 11 / 30

The Branson-Ørsted formula for space forms: the sphere

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 12 / 30

The Branson-Ørsted formula for space forms: the sphere

On S4 with the round metric, I[w] = 0,

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 12 / 30

The Branson-Ørsted formula for space forms: the sphere

On S4 with the round metric, I[w] = 0, II[w] =

• [(∆w)2 + 2|∇w|2] dv − 8π2 log

e4(w−w) dv

• (compare with the formula for S on the 2-sphere);
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 12 / 30

The Branson-Ørsted formula for space forms: the sphere

On S4 with the round metric, I[w] = 0, II[w] =

• [(∆w)2 + 2|∇w|2] dv − 8π2 log

e4(w−w) dv

• (compare with the formula for S on the 2-sphere);

III[w] = 12

• (∆w + |∇w|2)2 dv − 48
• |∇w|2 dv.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 12 / 30

The Branson-Ørsted formula for space forms: the sphere

On S4 with the round metric, I[w] = 0, II[w] =

• [(∆w)2 + 2|∇w|2] dv − 8π2 log

e4(w−w) dv

• (compare with the formula for S on the 2-sphere);

III[w] = 12

• (∆w + |∇w|2)2 dv − 48
• |∇w|2 dv.

Theorem (i) (Beckner, ’93) II[w] ≥ 0 with equality iff e2wg = ϕ∗g.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 12 / 30

The Branson-Ørsted formula for space forms: the sphere

On S4 with the round metric, I[w] = 0, II[w] =

• [(∆w)2 + 2|∇w|2] dv − 8π2 log

e4(w−w) dv

• (compare with the formula for S on the 2-sphere);

III[w] = 12

• (∆w + |∇w|2)2 dv − 48
• |∇w|2 dv.

Theorem (i) (Beckner, ’93) II[w] ≥ 0 with equality iff e2wg = ϕ∗g. (ii) (Beckner, Chang-Yang) III[w] ≥ 0 with equality iff e2wg = ϕ∗g.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 12 / 30

The coefficients γi.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 13 / 30

The coefficients γi.

1 Let A = ∆ − 1

6R, the conformal laplacian. In this case,

(γ1, γ2, γ3) = (1, −4, −2/3).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 13 / 30

The coefficients γi.

1 Let A = ∆ − 1

6R, the conformal laplacian. In this case,

(γ1, γ2, γ3) = (1, −4, −2/3).

2 A = P, the Paneitz operator:

Pgφ = (−∆)2φ + ∇j (2Rij − 2 3Rgij)∇iφ

• ,

where Rij are the components of the Ricci tensor. log det P was considered by Connes ’94 in connection with quantum gravity.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 13 / 30

The coefficients γi.

1 Let A = ∆ − 1

6R, the conformal laplacian. In this case,

(γ1, γ2, γ3) = (1, −4, −2/3).

2 A = P, the Paneitz operator:

Pgφ = (−∆)2φ + ∇j (2Rij − 2 3Rgij)∇iφ

• ,

where Rij are the components of the Ricci tensor. log det P was considered by Connes ’94 in connection with quantum gravity. Branson ’96 computed the coefficients in a separate paper, and found (γ1, γ2, γ3) = (−1/4, −14, 8/3).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 13 / 30

Another example: Cheeger’s half-torsion

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 14 / 30

Another example: Cheeger’s half-torsion

For (M2k, g) an even-dimensional manifold, define τh = (det(−∆0))n(det(−∆2))n−4 . . . (det(−∆1))n−2(det(−∆3))n−6 · · ·, where ∆p denotes the Hodge laplacian on p-forms.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 14 / 30

Another example: Cheeger’s half-torsion

For (M2k, g) an even-dimensional manifold, define τh = (det(−∆0))n(det(−∆2))n−4 . . . (det(−∆1))n−2(det(−∆3))n−6 · · ·, where ∆p denotes the Hodge laplacian on p-forms.

• Notice this only involves p for p < n/2. In four dimensions,

log τh = 4 log det(−∆0) − 2 log det(−∆1).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 14 / 30

Another example: Cheeger’s half-torsion

For (M2k, g) an even-dimensional manifold, define τh = (det(−∆0))n(det(−∆2))n−4 . . . (det(−∆1))n−2(det(−∆3))n−6 · · ·, where ∆p denotes the Hodge laplacian on p-forms.

• Notice this only involves p for p < n/2. In four dimensions,

log τh = 4 log det(−∆0) − 2 log det(−∆1).

• The half-torsion plays a role in self-dual field theory; cf. Witten ’97.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 14 / 30

Another example: Cheeger’s half-torsion

For (M2k, g) an even-dimensional manifold, define τh = (det(−∆0))n(det(−∆2))n−4 . . . (det(−∆1))n−2(det(−∆3))n−6 · · ·, where ∆p denotes the Hodge laplacian on p-forms.

• Notice this only involves p for p < n/2. In four dimensions,

log τh = 4 log det(−∆0) − 2 log det(−∆1).

• The half-torsion plays a role in self-dual field theory; cf. Witten ’97.
• In general, the Hodge laplacian is not conformally covariant. However,

the ratio above has the requisite conformal properties for deriving a Polyakov-type formula (cf. Juhl ’09). The coefficients for the corresponding functional are (γ1, γ2, γ3) = (−13, −248, 116/3).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 14 / 30

Key point: log det L is a convex combination of II and III; for log det P (and log τh), γ2 and γ3 have different signs.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 15 / 30

Key point: log det L is a convex combination of II and III; for log det P (and log τh), γ2 and γ3 have different signs.

• In particular, on S4 we have log det L ≤ 0, with equality only for the

round metric (modulo the conformal group). However....

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 15 / 30

Key point: log det L is a convex combination of II and III; for log det P (and log τh), γ2 and γ3 have different signs.

• In particular, on S4 we have log det L ≤ 0, with equality only for the

round metric (modulo the conformal group). However.... Proposition (G-Malchiodi) For any conformal four-manifold, inf FP = −∞, sup FP = +∞. Thus log det L and log det P exhibit quite different variational properties.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 15 / 30

A geometric detour: the geometry of the Euler equations

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 16 / 30

A geometric detour: the geometry of the Euler equations

• If we view the functionals I, II, III separately, what (geometric) property

does a critial metric satisfy?

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 16 / 30

A geometric detour: the geometry of the Euler equations

• If we view the functionals I, II, III separately, what (geometric) property

does a critial metric satisfy? I: ˆ g = e2wg is critical for I ⇔ ˆ g satisfies

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 16 / 30

A geometric detour: the geometry of the Euler equations

• If we view the functionals I, II, III separately, what (geometric) property

does a critial metric satisfy? I: ˆ g = e2wg is critical for I ⇔ ˆ g satisfies |W (ˆ g)|2 ≡ const, where W is the Weyl tensor.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 16 / 30

A geometric detour: the geometry of the Euler equations

• If we view the functionals I, II, III separately, what (geometric) property

does a critial metric satisfy? I: ˆ g = e2wg is critical for I ⇔ ˆ g satisfies |W (ˆ g)|2 ≡ const, where W is the Weyl tensor. III: ˆ g = e2wg is critical for I ⇔ ˆ g satisfies

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 16 / 30

A geometric detour: the geometry of the Euler equations

• If we view the functionals I, II, III separately, what (geometric) property

does a critial metric satisfy? I: ˆ g = e2wg is critical for I ⇔ ˆ g satisfies |W (ˆ g)|2 ≡ const, where W is the Weyl tensor. III: ˆ g = e2wg is critical for I ⇔ ˆ g satisfies (−∆ˆ

g)Rˆ g = 0,

where R is the scalar curvature. (Consequently, Rˆ

g ≡ const.)

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 16 / 30

The geometry of the Euler equations, cont.

II: ˆ g = e2wg is critical for II ⇔ ˆ g satisfies

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 17 / 30

The geometry of the Euler equations, cont.

II: ˆ g = e2wg is critical for II ⇔ ˆ g satisfies Qˆ

g ≡ const.

where Q is the Q-curvature:

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 17 / 30

The geometry of the Euler equations, cont.

II: ˆ g = e2wg is critical for II ⇔ ˆ g satisfies Qˆ

g ≡ const.

where Q is the Q-curvature: Q = 1 12

• − ∆R + R2 − 3|Ric|2

.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 17 / 30

The geometry of the Euler equations, cont.

II: ˆ g = e2wg is critical for II ⇔ ˆ g satisfies Qˆ

g ≡ const.

where Q is the Q-curvature: Q = 1 12

• − ∆R + R2 − 3|Ric|2

. Euler equation of FA ˆ g = e2wg is critical for FA ⇔ ˆ g satisfies γ1|W (ˆ g)|2 + γ2Qˆ

g + γ3(−∆ˆ g)Rˆ g ≡ const.

(E-L)

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 17 / 30

The geometry of the Euler equations, cont.

Some applications of the Euler equation:

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 18 / 30

The geometry of the Euler equations, cont.

Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions;

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 18 / 30

The geometry of the Euler equations, cont.

Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions; A conformally invariant sphere theorem. More precisely: sharp characterizations of S4, CP2, and S3 × S1 via L2-curvature estimates. (This uses Ricci flow and properties of the Euler equation);

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 18 / 30

The geometry of the Euler equations, cont.

Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions; A conformally invariant sphere theorem. More precisely: sharp characterizations of S4, CP2, and S3 × S1 via L2-curvature estimates. (This uses Ricci flow and properties of the Euler equation);

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 18 / 30

The geometry of the Euler equations, cont.

Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions; A conformally invariant sphere theorem. More precisely: sharp characterizations of S4, CP2, and S3 × S1 via L2-curvature estimates. (This uses Ricci flow and properties of the Euler equation); Note 4π2χ(M) = 1 2

• W 2 dV +
• Q dV .

Hence, the LHS of (E-L) integrates to a conformal invariant. (Compare with prescribing Gauss curvature in 2-d).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 18 / 30

Existence results for critical metrics for general 4-manifolds

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Existence results for critical metrics for general 4-manifolds

Theorem (Chang-Yang, ’95) Assume

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Existence results for critical metrics for general 4-manifolds

Theorem (Chang-Yang, ’95) Assume γ2 and γ3 have the same sign;

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Existence results for critical metrics for general 4-manifolds

Theorem (Chang-Yang, ’95) Assume γ2 and γ3 have the same sign; A smallness condition on the total Q-curvature (alternatively,

• |W |2dv ≤ c(χ(M4)));
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Existence results for critical metrics for general 4-manifolds

Theorem (Chang-Yang, ’95) Assume γ2 and γ3 have the same sign; A smallness condition on the total Q-curvature (alternatively,

• |W |2dv ≤ c(χ(M4)));

Then an extremal for F = γ1I + γ2II + γ3III exists.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Existence results for critical metrics for general 4-manifolds

Theorem (Chang-Yang, ’95) Assume γ2 and γ3 have the same sign; A smallness condition on the total Q-curvature (alternatively,

• |W |2dv ≤ c(χ(M4)));

Then an extremal for F = γ1I + γ2II + γ3III exists. Moreover, it is smooth (G-Chang-Yang, Viaclovsky-Uhlenbeck).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Existence results for critical metrics for general 4-manifolds

Theorem (Chang-Yang, ’95) Assume γ2 and γ3 have the same sign; A smallness condition on the total Q-curvature (alternatively,

• |W |2dv ≤ c(χ(M4)));

Then an extremal for F = γ1I + γ2II + γ3III exists. Moreover, it is smooth (G-Chang-Yang, Viaclovsky-Uhlenbeck).

• The “smallness” condition is related to the best constant in the

Moser-Trudinger inequality of Adams.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Existence results for critical metrics for general 4-manifolds

Theorem (Chang-Yang, ’95) Assume γ2 and γ3 have the same sign; A smallness condition on the total Q-curvature (alternatively,

• |W |2dv ≤ c(χ(M4)));

Then an extremal for F = γ1I + γ2II + γ3III exists. Moreover, it is smooth (G-Chang-Yang, Viaclovsky-Uhlenbeck).

• The “smallness” condition is related to the best constant in the

Moser-Trudinger inequality of Adams.

• This result does not apply to log det P (or log τh).
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 19 / 30

Back to the sphere

For (M, g) the standard 4-sphere,

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 20 / 30

Back to the sphere

For (M, g) the standard 4-sphere, FL[w] =

• S4
• − 12(∆w)2 − 16|∇w|2∆w − 8|∇w|4 + 24|∇w|2

dv + 32π2 log

S4 e4(w−w) dv

• ,
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 20 / 30

Back to the sphere

For (M, g) the standard 4-sphere, FL[w] =

• S4
• − 12(∆w)2 − 16|∇w|2∆w − 8|∇w|4 + 24|∇w|2

dv + 32π2 log

S4 e4(w−w) dv

• ,

while FP[w] =

• S4
• 18(∆w)2 + 64|∇w|2∆w + 32|∇w|4 − 60|∇w|2

dv + 112π2 log

S4 e4(w−w) dv

• .
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 20 / 30

Back to the sphere

For (M, g) the standard 4-sphere, FL[w] =

• S4
• − 12(∆w)2 − 16|∇w|2∆w − 8|∇w|4 + 24|∇w|2

dv + 32π2 log

S4 e4(w−w) dv

• ,

while FP[w] =

• S4
• 18(∆w)2 + 64|∇w|2∆w + 32|∇w|4 − 60|∇w|2

dv + 112π2 log

S4 e4(w−w) dv

• .
• Note: The cross-term (∆w)|∇w|2 compared to positive terms, and the

interaction of logarithmic term with derivative terms. This accounts for the lack of boundedness.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 20 / 30

FL = log det L on S4

Question Are there other critical points in the conformal class of the round metric?

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 21 / 30

FL = log det L on S4

Question Are there other critical points in the conformal class of the round metric?

• For log det L, the answer is no (G, ’97). However, this result is very

geometric, and based on curvature estimates. It doesn’t seem to exploit the variational structure at all.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 21 / 30

FL = log det L on S4

Question Are there other critical points in the conformal class of the round metric?

• For log det L, the answer is no (G, ’97). However, this result is very

geometric, and based on curvature estimates. It doesn’t seem to exploit the variational structure at all.

• Recall FP[0] = 0, and Branson showed w0 = 0 is a local minimizer.

(Same holds for log τh). As we noted above, it cannot be a global minimizer.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 21 / 30

FL = log det L on S4

Question Are there other critical points in the conformal class of the round metric?

• For log det L, the answer is no (G, ’97). However, this result is very

geometric, and based on curvature estimates. It doesn’t seem to exploit the variational structure at all.

• Recall FP[0] = 0, and Branson showed w0 = 0 is a local minimizer.

(Same holds for log τh). As we noted above, it cannot be a global minimizer.

• Branson’s calculation works on all space forms, and indicates a

Mountain-Pass structure. However...

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 21 / 30

FL = log det L on S4

Question Are there other critical points in the conformal class of the round metric?

• For log det L, the answer is no (G, ’97). However, this result is very

geometric, and based on curvature estimates. It doesn’t seem to exploit the variational structure at all.

• Recall FP[0] = 0, and Branson showed w0 = 0 is a local minimizer.

(Same holds for log τh). As we noted above, it cannot be a global minimizer.

• Branson’s calculation works on all space forms, and indicates a

Mountain-Pass structure. However... FP does not satisfy the Palais-Smale condition;

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 21 / 30

FL = log det L on S4

Question Are there other critical points in the conformal class of the round metric?

• For log det L, the answer is no (G, ’97). However, this result is very

geometric, and based on curvature estimates. It doesn’t seem to exploit the variational structure at all.

• Recall FP[0] = 0, and Branson showed w0 = 0 is a local minimizer.

(Same holds for log τh). As we noted above, it cannot be a global minimizer.

• Branson’s calculation works on all space forms, and indicates a

Mountain-Pass structure. However... FP does not satisfy the Palais-Smale condition; In fact, due to the poly-homogeneity, there are many ways in which a P-S sequence can diverge (i.e., not only ”bubbling”).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 21 / 30

log det P on S4, cont.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 22 / 30

log det P on S4, cont.

Theorem (G-Malchiodi, ’11) The 4-sphere admits a rotationally symmetric critical metric for log det P which is not conformally equivalent to the round metric. In particular, and in contrast to the case of log det L, uniqueness fails.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 22 / 30

log det P on S4, cont.

Theorem (G-Malchiodi, ’11) The 4-sphere admits a rotationally symmetric critical metric for log det P which is not conformally equivalent to the round metric. In particular, and in contrast to the case of log det L, uniqueness fails.

• Same result for log τh.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 22 / 30

log det P on S4, cont.

Theorem (G-Malchiodi, ’11) The 4-sphere admits a rotationally symmetric critical metric for log det P which is not conformally equivalent to the round metric. In particular, and in contrast to the case of log det L, uniqueness fails.

• Same result for log τh.
• We speculate that all constant curvature spaces admit second solutions.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 22 / 30

An overview of the proof

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 23 / 30

An overview of the proof

• The Euler equation for FP is

9∆2w + 32|∇2w|2 − 32(∆w)2 − 32∆u |∇u|2 − 32∇w, ∇|∇w|2 + 78∆u + 96|∇w|2 − 42 + 42e4w = 0.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 23 / 30

An overview of the proof

• The Euler equation for FP is

9∆2w + 32|∇2w|2 − 32(∆w)2 − 32∆u |∇u|2 − 32∇w, ∇|∇w|2 + 78∆u + 96|∇w|2 − 42 + 42e4w = 0.

• Since we are looking for rotationally symmetric solutions, we can use the

fact that S4 \ {N, P} is conformal to the cylinder R × S3 and ’lift’ the equation an ODE on the cylinder:

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 23 / 30

An overview of the proof

• The Euler equation for FP is

9∆2w + 32|∇2w|2 − 32(∆w)2 − 32∆u |∇u|2 − 32∇w, ∇|∇w|2 + 78∆u + 96|∇w|2 − 42 + 42e4w = 0.

• Since we are looking for rotationally symmetric solutions, we can use the

fact that S4 \ {N, P} is conformal to the cylinder R × S3 and ’lift’ the equation an ODE on the cylinder: 9u

′′′′ − 96u′′(u′)2 + 60u′′ + 42e4u = 0. (E)

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 23 / 30

An overview of the proof

• The Euler equation for FP is

9∆2w + 32|∇2w|2 − 32(∆w)2 − 32∆u |∇u|2 − 32∇w, ∇|∇w|2 + 78∆u + 96|∇w|2 − 42 + 42e4w = 0.

• Since we are looking for rotationally symmetric solutions, we can use the

fact that S4 \ {N, P} is conformal to the cylinder R × S3 and ’lift’ the equation an ODE on the cylinder: 9u

′′′′ − 96u′′(u′)2 + 60u′′ + 42e4u = 0. (E)

From the evenness of u we require the initial conditions u′(0) = 0, u′′′(0) = 0.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 23 / 30

The ODE

• We also need the asymptotic conditions as t → ∞:

u′′(t) → 0, u′(t) → −1 (so u descends to a solution on S4), t e4u ds → 2 3 (volume normalization)

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 24 / 30

The ODE

• We also need the asymptotic conditions as t → ∞:

u′′(t) → 0, u′(t) → −1 (so u descends to a solution on S4), t e4u ds → 2 3 (volume normalization)

• The solution u0(t) = − log cosh t represents the round metric.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 24 / 30

The ODE

• We also need the asymptotic conditions as t → ∞:

u′′(t) → 0, u′(t) → −1 (so u descends to a solution on S4), t e4u ds → 2 3 (volume normalization)

• The solution u0(t) = − log cosh t represents the round metric.
• As t → ∞, a solution with the right asymptotics should ’shadow’

solutions of 9u

′′′′ − 96u′′(u′)2 + 60u′′ = 0. (E∞)

This is integrable and has a 1-parameter family of solutions.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 24 / 30

A conservation law

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 25 / 30

A conservation law

Lemma Admissible solutions of (E) satisfy the the equation 9 4u′′′′ − 9u′u′′′ − 24u′′(u′)2 + 9 2(u′′)2 + 15u′′ + 24(u′)4 − 30(u′)2 + 6 = 0.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 25 / 30

A conservation law

Lemma Admissible solutions of (E) satisfy the the equation 9 4u′′′′ − 9u′u′′′ − 24u′′(u′)2 + 9 2(u′′)2 + 15u′′ + 24(u′)4 − 30(u′)2 + 6 = 0. Also, −9 2[u′′(0)]2 + 21 2 e4u(0) = 6, so that the initial conditions are completely determined by u′′(0).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 25 / 30

A conservation law

Lemma Admissible solutions of (E) satisfy the the equation 9 4u′′′′ − 9u′u′′′ − 24u′′(u′)2 + 9 2(u′′)2 + 15u′′ + 24(u′)4 − 30(u′)2 + 6 = 0. Also, −9 2[u′′(0)]2 + 21 2 e4u(0) = 6, so that the initial conditions are completely determined by u′′(0).

• The conservation law above reduces the equation to a third order

autonmous equation in u′.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 25 / 30

The system

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 26 / 30

The system

Let x = −u′, y = −u′′, z = −u′′′,

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 26 / 30

The system

Let x = −u′, y = −u′′, z = −u′′′, then the third order autonomous equation becomes the system (A):    x′ = y, y′ = z, z′ =

32 3 (x − 1)

• x − 1

2

• (x + 1)
• x + 1

2

• − 4xz + 2y2 + 32

3 x2y − 20 3 y,

with initial conditions    x(0) = 0, y(0) = −u′′(0), z(0) = 0.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 26 / 30

The system, cont.

• Key point:

Lemma The system (A) contains solutions of both (E) and (E∞).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 27 / 30

The system, cont.

• Key point:

Lemma The system (A) contains solutions of both (E) and (E∞).

• This (surprising!) fact alllows us to study the asymptotics of the former

by comparing with the (explicit) behavior of the latter.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 27 / 30

The system, cont.

• Key point:

Lemma The system (A) contains solutions of both (E) and (E∞).

• This (surprising!) fact alllows us to study the asymptotics of the former

by comparing with the (explicit) behavior of the latter.

• Goal: Goal Look for solutions of the system starting from the y-axis

(i.e., points where u′ = 0 and u′′′ = 0) which converge asymptotically to the point (1, 0, 0).

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 27 / 30

The proof in pictures

! ! ! !

• !
• A 1-parameter family of periodic solutions sweeps out a ’disk’ D.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 28 / 30

The proof in pictures, cont.

• If our initial condition is

X(0) = (0, 1, 0) we get the solution corresponding to the round metric.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 29 / 30

The proof in pictures, cont.

• If our initial condition is

X(0) = (0, 1, 0) we get the solution corresponding to the round metric. Key Proposition For ǫ > 0 small, let Xǫ denote the solution of the system with initial condition

• Xǫ(0) = (0, 1 − ǫ, 0).
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 29 / 30

The proof in pictures, cont.

• If our initial condition is

X(0) = (0, 1, 0) we get the solution corresponding to the round metric. Key Proposition For ǫ > 0 small, let Xǫ denote the solution of the system with initial condition

• Xǫ(0) = (0, 1 − ǫ, 0).

(i) If ǫ > 0 is small enough, then Xǫ(t) exists for all time and approaches

• ne of the periodic solutions which lives in the ’disk’ D.
• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 29 / 30

The proof in pictures, cont.

• If our initial condition is

X(0) = (0, 1, 0) we get the solution corresponding to the round metric. Key Proposition For ǫ > 0 small, let Xǫ denote the solution of the system with initial condition

• Xǫ(0) = (0, 1 − ǫ, 0).

(i) If ǫ > 0 is small enough, then Xǫ(t) exists for all time and approaches

• ne of the periodic solutions which lives in the ’disk’ D.

(ii) For ǫ > 0 large enough, Xǫ blows up in finite time.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 29 / 30

The proof in pictures, cont.

• If our initial condition is

X(0) = (0, 1, 0) we get the solution corresponding to the round metric. Key Proposition For ǫ > 0 small, let Xǫ denote the solution of the system with initial condition

• Xǫ(0) = (0, 1 − ǫ, 0).

(i) If ǫ > 0 is small enough, then Xǫ(t) exists for all time and approaches

• ne of the periodic solutions which lives in the ’disk’ D.

(ii) For ǫ > 0 large enough, Xǫ blows up in finite time.

• Strategy: Show that we can ’hit the sweet spot’; i.e., there exists an

ǫ0 > 0 such that Xǫ0 exists for all time and approaches (1, 0, 0) ∈ ∂D.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 29 / 30

The proof in pictures, cont.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 30 / 30

The proof in pictures, cont.

• M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations )

Determinants and CI-operators June 7, 2012 30 / 30