Eigenvalue bounds in CR and Quaternionic Contact geometries under - - PowerPoint PPT Presentation

Eigenvalue bounds in CR and Quaternionic Contact geometries under positive ”Ricci” bound

Dimiter Vassilev, University of New Mexico . . Collaborators: Stefan Ivanov (University of Sofia) & Alexander Petkov, (University of Sofia) September 30, 2014

Comparison results

Suppose (M, h) complete Riemannian manifold of dimension n with Ric(X, X) ≥ (n − 1)h(X, X). △f = λf. i) λ ≥ n. ii) If ∇2f = −fh, f ≡ 0, then (M, h) is isometric with Sn(1). Notes:

• 1. Bonnet-Myers: M is compact, diam (M) ≤ π and π1(M) is

finite.

• 2. Lichnerowicz: λ ≥ n, Bochner-Weitzenb¨
• ck formula.
• 3. Obata: if there is λ = n, then (M, h) is isometric with Sn(1).
• 4. S.Y. Cheng ’75 (improved Toponogov): diam (M) = π iff M

is isometric to Sn(1).

Lichnerowicz’ estimate

Bochner’s identity (△ ≥ 0): −1 2△|∇f|2 = |∇df|2 − h(∇(△f), ∇f) + Ric(∇f, ∇f). Therefore 0 =

• M |(∇df)0|2 + 1

n(△f)2 − h(∇(△f), ∇f) + Ric(∇f, ∇f) dvol.

For △f = λf we get 0 =

• M

|(∇df)0|2 + 1 nλ|∇f|2 − λ|∇f|2 + Ric(∇f, ∇f) dvol =

• M

|(∇df)0|2 dvol +

• M

Ric(∇f, ∇f) − n − 1 n λ|∇f|2 dvol ≥

• M

|(∇df)0|2 dvol + n − 1 n

• M

(n − λ)|∇f|2 dvol for Ric(∇f, ∇f) ≥ (n − 1)|∇f|2. Hence 0 ≥ n − λ.

Equality implies Einstein

Proposition

If (M, h) is a compact Riemannian manifold of dimension n with Ric(X, X) ≥ (n − 1)h(X, X) and △f = nf, then M is Einstein. Key:

◮ ∇Ric0(∇f, X, Y) = 2fRic0(X, Y) + trace term, hence

L∇f|Ric0|2k = 4kf|Ric0|2k.

◮ M |Ric0|2kf 2 dvol = 1 n

• M h(∇|Ric0|2kf, ∇f) dvol

= 1

n

• M |Ric0|2k|∇f|2 dvol + 4k

n

• M |Ric0|2kf 2 dvol.

◮ (n − 4k)

• M |Ric0|2kf 2 dvol =
• M |Ric0|2k|∇f|2 dvol, hence

choosing k > n/4 it follows Ric0 = 0.

sub-Riemannian results

• 1. Rumin, M., ’94: Bonnet-Myers type theorem on general

3-D CR manifolds.

• 2. Hughen, K., ’95: Bonnet-Myers type theorem on 3-D

Sasakian.

• 3. Chanillo, S., Yang, P

.-C.: Isoperimetric inequalities & volume comparison theorems on CR manifolds. Ann. Sc.

• Norm. Super. Pisa, (2009)
• 4. (Bakry-Emery) F

. Baudoin, N. Garofalo, I. Munive, B. Kim,

• J. Wang; E. Grong & A. Thalmaier: curvature-dimension

inequalities ⇒ Myers-type theorems, volume doubling, Li-Yau inequality, Sobolev, Harnack,...

• 5. (Sturm, Lott & Villani) A. Agrachev, P

. Lee, Chengbo Li, I. Zelenko, D. Barilari & L. Rizzi,...: Bishop comparison theorem, Harnack, Laplacian/ Hessian comparison,..

• 6. R. Hladky - Lichnerowicz type estimates, Bonnet-Myers,..

CR SETTING

CR manifolds

Definition (SPCSH manifold)

(M, θ, J) is strictly pseudoconvex pseudohermitian manifold if θ is a contact form, H = ker θ has a compatible Hermitian structure: J : H → H, J2 = −idH, 2g(X, Y) def = dθ(X, JY) is positive definite on H; g(X, Y) = g(JX, JY); [JX, JY] − [X, Y] − J[JX, Y] − J[X, JY] = 0. Reeb field ξ: θ(ξ) = 1 and ξdθ = 0. 2-form: ω(X, Y) def = g(JX, Y).

Theorem (Tanaka-Webster connection)

(i) ∇ξ = ∇J = ∇θ = ∇g = 0; (ii) the torsion T(A, B) = ∇AB − ∇BA − [A, B] satisfies: T(X, Y) = 2ω(X, Y)ξ and T(ξ, X) ∈ H, g(T(ξ, X), Y) = g(T(ξ, Y), X) = −g(T(ξ, JX), JY). The Webster torsion A, A def = T(ξ, .) : H → H, is a symmetric (2, 0) + (0, 2) tensor. A is the obstruction for a pseudohermitian manifold to be Sasakian.

Curvature of the Tanaka-Webster connection

Define the Riemannian metric ”h = g + η2”. Let {ǫa}2n

a=1-ONB of

the horizontal space H.

• 1. Tanaka-Webster curvature:

R(A, B)C def = [∇A, ∇B]C − ∇[A,B]C and R(A, B, C, D) def = h(R(A, B)C, D).

• 2. Ricci tensor:

Ric(A, B) = R(ǫa, A, B, ǫa) def = 2n

a=1 R(ǫa, A, B, ǫa); scalar

curvature S = Ric(ǫa, ǫa);

• 3. Ricci form: ρ(A, B) = 1

2 R(A, B, ǫa, Jǫa).

Type decomposition of the Ricci tensor: Ric(X, Y) = ρ(JX, Y) + 2(n − 1)A(JX, Y). Ricci identity example: ∇3f(X, Y, ξ) − ∇3f(ξ, X, Y) = ∇2f(AX, Y) + ∇2f(X, AY) + (∇XA)(Y, ∇f) + (∇YA)(X, ∇f) − (∇∇f)A(X, Y). Horizontal divergence e.g.’s: sub-Laplacian: △f = −∇2f(ǫa, ǫa); CR contracted 2nd Bianchi:

CR Lichnerowicz theorem

Theorem (Greenleaf, A. ’85) for n ≥ 3; Li, S.-Y., & Luk, H.-S. ’04 for n=2)

Let M be a compact spcph manifold of dimension 2n + 1, s.t., for some k0 = const > 0 we have the Lichnerowicz-type bound Ric(X, X) + 4A(X, JX) ≥ k0g(X, X), X ∈ H. If n > 1, then any eigenvalue λ of the sub-Laplacian satisfies λ ≥

n n+1k0.

The standard Sasakian unit sphere has first eigenvalue equal to 2n with eigenspace spanned by the restrictions of all linear functions to the sphere.

Theorem (Chiu, H.-L. ’06)

If n = 1 the estimate λ ≥

n n+1k0 holds assuming in addition that

the CR-Paneitz operator is non-negative

• M f · Cf Volθ ≥ 0,

where Cf is the CR-Paneitz operator, Cf = ∇4f(ea, ea, eb, eb) + ∇4f(ea, Jea, eb, Jeb) − 4n∇∗A(J∇f) − 4n g(∇2f, JA). Note: Li, S.-Y., & Luk, H.-S. ’04 for n = 1 with condition. Given a function f we define the one form, Pf(X) = ∇3f(X, eb, eb) + ∇3f(JX, eb, Jeb) + 4nA(X, J∇f) so we have Cf = −∇∗P.

The CR-Paneitz operator

The divergence formula turns the non-negativity condition ”C ≥ 0” into

• M

f · Cf Volθ = −

• M

Pf(∇f) Volθ ≥ 0.

◮ For n > 1 always C ≥ 0, Graham,C.R., & Lee, J.M ’88. ◮ In the three dimensional case C ≥ 0 is a CR invariant by

the pseudo-conformal invariance of C, Hirachi ’93, if ˆ θ = φ2θ then ˆ C = φ−4C.

Embedded CR and the CR-Paneitz operator

◮ If A = 0, then C = const✷b ✷b ≥ 0, where ✷b is the Kohn

• Laplacian. Furthermore, M is embeddable Lempert, L. ’92.

◮ If n = 1, C ≥ 0 and S > 0 (for e.g. the CR Yamabe

constant is positive), then M can be globally embedded into CN for some N. Chanillo, S., Chiu, H.-L., Yang, P . ’12.

Proof of the CR Lichnerowicz estimate

CR Bochner idenity: −1 2△|∇f|2 = |∇df|2−g(∇(△f), ∇f)+Ric(∇f, ∇f)+2A(J∇f, ∇f) + 4∇df(ξ, J∇f). The last term can be related to the traces of ∇2f:

• M

∇2f(ξ, J∇f) Volθ = −

• M

1 2ng(∇2f, ω)2 + A(J∇f, ∇f) Volθ and also using the Paneitz operator

• M

∇2f(ξ, J∇f) Volθ =

• M

− 1 2n (△f)2+A(J∇f, ∇f)− 1 2nPf(∇f) Volθ.

”key” from the CR Bochner identity

Integrating the CR Bochner idenity (for arbitrary function f) and using the last two formulas for the

• M ∇2f(ξ, J∇f) Volθ term we

find 0 =

• M

Ric(∇f, ∇f) + 4A(J∇f, ∇f) − n + 1 n (△f)2 Volθ +

• M
• (∇2f)
• 2

− 1 2n(△f)2 − 1 2ng(∇2f, ω)2 Volθ +

• M
• − 3

2nP(∇f)

• Volθ.

Notice that

• 1

√ 2ng, 1 √ 2nω

• is an orthonormal set in the (1, 1)

space with non-zero traces, so

• (∇2f)[0]
• 2 def

=

• (∇2f)
• 2

− 1 2n(△f)2 − 1 2ng(∇2f, ω)2.

Assuming △f = λf and the ”Ricci” bound we obtain the inequality: 0 ≥

• M
• k0 − n + 1

n λ

• |∇f|2 Volθ +
• M
• (∇2f)[0]
• 2 Volθ

− 3 2n

• M

Pf(∇f) Volθ, which implies λ ≥

n n+1k0 with equality holding iff

∇2f = 1 2n(△f) · g + 1 2ng(∇2f, ω) · ω and

• M Pf(∇f) Volθ = 0 (use the extra assumption for n = 1!).

CR Obata type theorem

Theorem (n ≥ 2, Li, S.-Y., Wang, X. ’13; n=1 w/ Ivanov ’14)

Suppose (M, J, θ), dim M = 2n + 1, is a compact spcph manifold which satisfies the Lichnerowicz-type bound. If n ≥ 2, then λ =

n n+1k0 is an eigenvalue iff up-to a scaling (M, J, θ) is

the standard pseudo-Hermitian CR structure on the unit sphere in Cn+1. If n = 1 the same conclusion holds assuming in addition C ≥ 0. Earlier results

◮ Sasakian case, Chang, S.-C., & Chiu, H.-L., for n ≥ 2 in J.

• Geom. Anal. ’09; for n = 1 in Math. Ann. ’09.

◮ Non-Sasakian case, Chang, S.-C., & Wu, C.-T., ’12,

assuming: (i) for n ≥ 2, Aαβ, ¯

β = 0 and Aαβ, γ¯ γ = 0;

(ii) for n = 1, A11, ¯

1 = 0 and P1f = 0. ◮ w/ S. Ivanov ’12 - assuming ∇∗A = 0 and C ≥ 0 when

n = 1.

QUATERNIONIC CONTACT SETTING

Quaternionic Contact Structures

Definition (O. Biquard ’99)

M4n+3-quaternionic contact if we have i) co-dim three distribution H, locally, H = 3

s=1 Ker ηs,

ηs ∈ T ∗M. ii) a 2-sphere bundle of ”almost complex structures” locally generated by Is : H → H, I2

s = −1, satisfying

I1I2 = −I2I1 = I3; iii) a ”metric” tensor g on H, s.t., g(IsX, IsY) = g(X, Y), and 2ωs(X, Y) ≡ 2g(IsX, Y) = dηs(X, Y), for all X, Y ∈ H.

Theorem (O. Biquard)

Under the above conditions and n > 1, there exists a unique subbundle V, s.t., TM = H ⊕ V and a linear connection ∇ on M with the properties: (i) V and H are parallel; (ii) g and Ω = 3

j=1 ωj ∧ ωj are parallel;

(iii) the torsion satisfies

◮ ∀X, Y ∈ H,

T(X, Y) = −[X, Y]|V ∈ V

◮ ∀ξ ∈ V, X ∈ H, Tξ(X) ≡ T(ξ, X) ∈ H and

Tξ ∈ (sp(n) + sp(1))⊥.

Reeb vector fields

◮ Note: V is generated by the Reeb vector fields {ξ1, ξ2, ξ3}

ηs(ξk) = δsk, (ξsdηs)|H = 0, (ξsdηk)|H = −(ξkdηs)|H.

◮ If the dimension of M is seven, n = 1, Reeb vector fields

might not exist.

◮ D. Duchemin showed that if we assume their existence,

then there is a connection as before. Henceforth, by a qc structure in dimension 7 we mean a qc structure satisfying the Reeb conditions

◮ We extend the horizontal metric g to a Riemannian metric

h on M by requiring span{ξ1, ξ2, ξ3} = V ⊥ H and h(ξs, ξt) = δst, h|H = g, h|V = η1 ⊗ η1 + η2 ⊗ η2 + η3 ⊗ η3, h(ξs, X) = 0. N.B. h as well as the Biquard connection do not depend on the action of SO(3) on V.

Quaternionic Heisenberg Group

G (H) = Hn × ImH, (q, ω) ∈ G (H), (qo, ωo) ◦ (q, ω) = (qo + q, ω + ωo + 2 Im qo ¯ q), i) ˜ Θ = (˜ Θ1, ˜ Θ2, ˜ Θ3) =

1 2 (dω − q · d¯

q + dq · ¯ q) or ˜ Θ1 = 1 2 dx − xαdtα + tαdxα − zαdyα + y αdzα ˜ Θ2 = 1 2 dy − yαdtα + zαdxα + tαdy α − xαdzα ˜ Θ2 = 1 2 dz − zαdtα − y αdxα + xαdyα + tαdzα. ii) Left-invariant horizontal vector fields Tα = ∂ ∂tα + 2xα ∂ ∂x + 2y α ∂ ∂y + 2zα ∂ ∂z , Xα = ∂ ∂xα − 2tα ∂ ∂x − 2zα ∂ ∂y + 2yα ∂ ∂z , Yα = ∂ ∂yα + 2zα ∂ ∂x − 2tα ∂ ∂y − 2xα ∂ ∂z , Zα = ∂ ∂zα − 2yα ∂ ∂x + 2xα ∂ ∂y − 2tα ∂ ∂z . iii) Left-invariant Reeb fields ξ1, ξ2, ξ3 are ξ1 = 2 ∂

∂x ,

ξ2 = 2 ∂

∂y ,

ξ3 = 2 ∂

∂z .

iv) On G (H), the left-invariant connection is the Biquard connection. It is flat!

Standard qc-structure on 3-Sasakain sphere

◮ Contact 3-form on the sphere S = {|q|2 + |p|2 = 1} ⊂ Hn × H,

˜ η = dq · ¯ q + dp · ¯ p − q · d¯ q − p · d¯ p.

◮ Identify G (H) with the boundary Σ of a Siegel domain in Hn × H,

Σ = {(q′, p′) ∈ Hn × H : Re p′ = |q′|2}, by using the map (q′, ω′) → (q′, |q′|2 − ω′).

◮ Cayley transform, C : S \ {pt.} → Σ,

(q′, p′) = C

• (q, p)
• = ((1 + p)−1 q, (1 + p)−1 (1 − p)).

◮ C∗ ˜

Θ =

1 2 |1+p |2 λ ˜

η ¯ λ, λ-unit quaternion (eg. of conformal quaternionic contact map).

◮ qc-curvature: R(A, B)C = [∇A, ∇B]C − ∇[A,B]C ◮ qc-Ricci tensor: Ric(A, B) = R(ea, A, B, ea)

def

= 4n

a=1 h(R(ea, A)B, ea);

◮ qc-scalar curvature: Scal = trH Ric = Ric(ea, ea); ◮ ”K¨

ahler” forms 2ωi(X, Y) = dηi(X, Y) = 2g(IiX, Y), ξωi = 0, ξ ∈ V; The Torsion Tensor (O. Biquard): Recall Tξ : H → H. Biquard showed: Tξj = T 0

ξj + IjU, U ∈ Ψ[3]. T 0 ξj -symmetric, IjU-skew-symmetric.

Theorem (w/ S. Ivanov, I. Minchev)

With T 0 def = T 0

ξ1 I1 + T 0 ξ2 I2 + T 0 ξ3 I3 ∈ Ψ[−1],

Ric = (2n + 2)T 0 + (4n + 10)U + Scal

4n g.

Definition

M is called qc-Einstein if T 0 = 0 and U = 0. M is called qc-pseudo-Einstein if U = 0.

Theorem (w/ S.Ivanov& I.Minchev, ’14 & arXiv:1306.0474)

Suppose Scal > 0. The next conditions are equivalent: i) (M4n+3, η) is qc-Einstein manifold. ii) M is locally 3-Sasakian: locally there exists a matrix Ψ ∈ C∞(U : SO(3)), s.t., Ψ · η is 3-Sasakian;

Volume and divergence formula

• 1. For a fixed local 1-form η and a fixed s ∈ {1, 2, 3} the form

Volη = η1 ∧ η2 ∧ η3 ∧ ω2n

s is a locally defined volume form.

• 2. In fact, Volη is independent of s and the local one forms η1, η2, η3 and

therefore it is a globally defined volume form denoted with Volη.

• 3. The (horizontal) divergence of a horizontal vector field/one-form

σ ∈ Λ1 (H) defined by ∇∗ σ = −tr|H∇σ = −∇σ(ea, ea) supplies the ”integration by parts” over compact M formula.

• M

(∇∗σ) Volη = 0.

QC Lichnerowicz

Theorem (w/ Ivanov, S., & Petkov, A. ’13 & ’14)

Let (M, η) be a compact QC manifold of dimension 4n + 3. Suppose, for αn = 2(2n+3)

2n+1 ,

βn = 4(2n−1)(n+2)

(2n+1)(n−1) and for any X ∈ H

L(X, X)

def

= 2Sg(X, X) + αnT 0(X, X) + βnU(X, X) ≥ 4g(X, X). If n = 1, assume in addition the positivity of the P-function of any

• eigenfunction. Then, any eigenvalue λ of the sub-Laplacian △ satisfies the

inequality λ ≥ 4n The 3-Sasakian sphere achieves equality in the Theorem. The eigenspace of the first non-zero eigenvalue of the sub-Laplacian on the unit 3-Sasakian sphere in Euclidean space is given by the restrictions to the sphere of all linear functions.

The QC Obata type theorem in the compact case

Theorem (w/ S. Ivanov & A. Petkov, arxiv1303.0409)

Let (M, η) be a compact QC manifold of dimension 4n + 3 which satisfies a Lichnerowicz’ type bound L(X, X) ≥ 4g(X, X). Then, there is a function f with △f = 4nf. if and only if:

◮ if n > 1, then M is qc-homothetic to the 3-Sasakian sphere; ◮ if n = 1, and M is qc-Einstein, i.e., T 0 = 0, then M is qc-homothetic to

the 3-Sasakian sphere. Notes:

• 1. The 7-D case is still open in the general case.
• 2. As we shall see later, the results follow from another theorem where
• nly completeness and knowledge of the horizontal Hessian are

assumed.

Definition of the QC P-function

a) The P−form of a function f is the 1-form Pf(X) = ∇3f(X, eb, eb) +

3

• t=1

∇3f(ItX, eb, Iteb) − 4nSdf(X) + 4nT 0(X, ∇f) − 8n(n − 2) n − 1 U(X, ∇f). b) The P−function of f is the function Pf(∇f). c) The C−operator is the 4-th order differential operator on M (independent

• f f!)

f → Cf = −∇∗Pf = (∇eaPf) (ea). d) The P−function of f is non-negative if

• M

f · Cf Volη = −

• M

Pf(∇f) Volη ≥ 0. If the above holds for any f ∈ C∞

• (M) we say that the C−operator is

non-negative, C ≥ 0.

Properties of the C-operator

Theorem (w/ S. Ivanov & A. Petkov, ’13)

C ≥ 0 for n > 1. Furthermore Cf = 0 iff (∇2f)[3][0](X, Y) = 0. In this case the P−form of f vanishes as well. The Sp(n)Sp(1)-invariant decomposition of ∇2f: (∇2f)[3](X, Y) = 1 4

• ∇2f(X, Y) +

3

• s=1

∇2f(IsX, IsY)

• .

(∇2f)[−1](X, Y) = 1 4

• 3∇2f(X, Y) −

3

• s=1

∇2f(IsX, IsY)

• .

Key: (∇ea(∇2f)[3][0])(ea, X) = n−1

4n Pf(X), hence

n − 1 4n

• M

f · Cf Volη = −n − 1 4n

• M

Pf(∇f) Volη =

• M

|(∇2f)[3][0]|2 Volη, after using the Ricci identities, the divergence formula and the orthogonality

• f the components of the horizontal Hessian.

The 7-D case

Theorem (w/ S. Ivanov & A. Petkov, ’13)

On a 7-D compact qc-Einstein manifold with Scal ≥ 0 the P−function of an eigenfunction of the sub-Laplacian is non-negative, i.e., △f = λf ⇒ −

• M

Pf(∇f) Volη ≥ 0. Keys:

◮ qc-Einstein ⇒ Scal = const; non-trivial in 7-D, use W qc [w/ Ivanov ’10]! ◮ qc-Einstein ⇒ ∇3f(ξs, X, Y) = ∇3f(X, Y, ξs); ◮ vertical space is integrable; ∇2f(ξk, ξj) − ∇2f(ξj, ξk) = −Sdf(ξi) ◮

M |Pf|2 Volθ = −(λ + 4S)

• M Pf(∇f) Volθ

Proof of Lichnerowicz’ type theorem

◮ QC Bochner identity; with Rf = 3

s=1 ∇2f(ξs, Is∇f) we have

− 1 2△|∇f|2 = |∇2f|2 − g (∇(△f), ∇f) + 2(n + 2)S|∇f|2 + 2(n + 2)T 0(∇f, ∇f) + 2(2n + 2)U(∇f, ∇f) + 4Rf.

◮

M RfVolη =

• M − 1

4nPn(∇f) − 1 4n(△f)2 − S|∇f|2Volη +

• M

n+1 n−1U(∇f, ∇f) Volη.

◮ using Ricci’s identities g(∇2f, ωs)

def

= ∇2f(ea, Isea) = −4ndf(ξs),

• M

RfVolη = −

• M

1 4n

3

• s=1

g(∇2f, ωs)2 + T 0(∇f, ∇f) − 3U(∇f, ∇f) Volη.

End of the proof of ”Lichnerowicz’” estimate

A substitution of a linear combination of the last two identities in the QC Bochner identity gives 0 =

• M

|∇2f|2 − 1 4n

• (△f)2 +

3

• s=1

[g(∇2f, ωs)]2 − 3 4nPn(∇f)Volη + 2n + 1 2

• M

L(∇f, ∇f) − λ n |∇f|2 Volη. With the Lichnerowicz type assumption, L(∇f, ∇f) ≥ 4|∇f|2, it follows ≥

• M

|(∇2f)0|2 − 3 4nPn(∇f)Volη + 2n + 1 2n

• M
• 4n − λ
• |∇f|2Volη.

For n = 1, when U = 0 trivially, remove formally the torsion tensor U terms - the formulas are still correct.

The QC Obata type theorem w/out compactness

Theorem (IPV arxiv1303.0409)

Let (M, η) be a quaternionic contact manifold of dimension 4n + 3 > 7 which is complete with respect to the associated Riemannian metric h = g + (η1)2 + (η2)2 + (η3)2.There exists a smooth f ≡const, s.t, ∇df(X, Y) = −fg(X, Y) −

3

• s=1

df(ξs)ωs(X, Y). if and only if the qc manifold (M, η, g, Q) is qc homothetic to the unit 3-Sasakian sphere.

Outline of proof of QC Obata

Part 1: show T 0 = 0 and U = 0, i.e., M is qc-Einstein.

• 1. find the remaining parts of the Hessian (w.r.t. the Biquard connection) in

terms of the torsion tensors.

• 2. A simple argument shows that T 0(Is∇f, ∇f) = U(Is∇f, ∇f) = 0.
• 3. Using the [−1]-component of the curvature tensor it follows

T 0(Is∇f, It∇f) = 0, s, t ∈ {1, 2, 3}, s = t.

• 4. Determine the torsion tensors T 0 and U in terms of ∇f and the tensor

U(∇f, ∇f). For e.g., |∇f|4T 0(X, Y) = − 2n

n−1U(∇f, ∇f)

• 3df(X)df(Y)−3

s=1 df(IsX)df(IsY)

• .
• 5. Formulas of the same type for ∇T 0 and ∇U.

N.B. In particular: (∇∇fU)(X, Y) = 2(n−1)

n+2 fU(X, Y), hence

L∇f|U|2 = 4(n−1)

n+2 f|U|2 as in the Riemannian case for Ric0!. Hence, in

the compact case we can use the Riemannian argument.

• 6. Thus, the crux of the matter is the proof that U(∇f, ∇f) = 0 (or

T 0(∇f, ∇f) = 0). This fact is achieved with the help of the Ricci identities, the contracted Bianchi second identity and many properties

• f the torsion of a qc-manifolds:

0 = ∇3f(ξi, Ii∇f, ∇f) − ∇3f(Ii∇f, ∇f, ξi) =

2 n+2fU(∇f, ∇f).

QC Obata cont’d

• 7. On a qc manifold with n > 1, the ”horizontal Hessian eq’n” implies that f

satisfies an elliptic PDE △hf = (4n+3)f+ n + 1 n(2n + 1)(∇eaT 0)(ea, ∇f)+ 3 (2n + 1)(n − 1)(∇eaU)(ea, ∇f). Part 2: The case of a qc-Einstein structure.

• 1. Show that (∇h)2f(X, Y) = −fh(X, Y), (h- Riemannian metric!).
• 2. from Obata’s result (M, h) is homothetic to the unit sphere in quaternion

space.

• 3. show qc-conformal flatness: use

Rh(A, B, C, D) = h(B, C)h(A, D) − h(B, D)h(A, C) and the relation between Rh and R, and then the formula for W qc(X, Y, Z, V).

• 4. (M, g, η, Q) is qc-conformal to S4n+3, i.e., we have η = κΨF ∗˜

η for some diffeomorphism F : M → S4n+3, 0 < κ ∈ C∞(M), and Ψ ∈ C∞(M : SO(3)) Use the qc-Liouville theorem.

Theorem ( ˇ Cap, A., & Slov´ ak, J., ’09; w/ Ivanov, S., & Petkov, A., arXiv:1303.0409)

Every qc-conformal transformation between open subsets of the 3-Sasakian unit sphere is the restriction of a global qc-conformal transformation. Rmrk: Cowling, M., & Ottazzi, A., Conformal maps of Carnot groups, arXiv:1312.6423.

• 5. compare the metrics on H to see homothety.

QC Conformal Curvature tensor

◮ ”Schouten” tensor L(X, Y) = 1

2T 0(X, Y) + U(X, Y) + Scal 32n(n+2) g(X, Y).

◮ Conformal curvature

W qc(X, Y, Z, V) = R(X, Y, Z, V) + (g L)(X, Y, Z, V) +

3

• s=1

(ωs IsL)(X, Y, Z, V) − 1 2

• (i,j,k)

ωi(X, Y)

• L(Z, IiV) − L(IiZ, V) + L(IjZ, IkV) − L(IkZ, IjV)
• −

3

• s=1

ωs(Z, V)

• L(X, IsY)−L(IsX, Y)
• + 1

2n(trL)

3

• s=1

ωs(X, Y)ωs(Z, V), where

(i,j,k) denotes the cyclic sum.

Conformal Flatness

Theorem (w/ Ivanov, S., ’10)

a)W qc is qc-conformal invariant, i.e., if ¯ η = κΨη then W qc

¯ η = φ W qc η ,

0 < κ ∈ C∞(M), and Ψ ∈ C∞(M : SO(3)). b) A qc-structure is locally qc-conformal to the standard flat qc-structure on the quaternionic Heisenberg group G (H) iff W qc = 0.

Corrolary

A qc manifold is locally quaternionic contact conformal to the quaternionic sphere S4n+3 if and only if the qc conformal curvature vanishes, W qc = 0.

Idea of Proof of the CR Obata theorem

◮ ∇3f(X, Y, ξ) = −df(ξ)g(X, Y) − (ξ2f)ω(X, Y) − 2fA(X, Y) +

∇A(X, Y, ∇f) + ∇A(Y, X, ∇f) − ∇A(∇f, X, Y).

◮ Obtain a formula for R(X, Y, Z, ∇f), Ric(X, ∇f) and Ric(JX, J∇f).

N.B.: If n = 1, Ric(X, Y) = Ric(JX, JY), hence the last two identities coincide.

◮ If n > 1, also using the (2, 0) + (0, 2) part of R(., ., X, Y) ⇒

∇2f(Y, ξ) = df(JY) + 2A(Y, ∇f). (*)

◮ For n > 1, |∇f|2A(Y, Z) = df(Y)A(∇f, Z) − df(JY)A(∇f, JZ). Hence,

|∇f|2|A|2 = 2|A∇f|2.

◮ From (*), ∇3f(X, Y, ξ) = −df(ξ)g(X, Y) + fω(X, Y) − 2fA(X, Y) −

2df(ξ)A(JX, Y) + 2∇A(X, Y, ∇f).

◮ For n > 1, comparing the two formulas for ∇3f(X, Y, ξ) it follows

∇A(∇f, X, Y) = 2df(ξ)A(JX, Y) (N.B. hence L∇f|A|2 = 0) and ∇2f(ξ, ξ) = −f + 1

n(∇∗A)(J∇f).

The ’missing” equation when n = 1

Proposition (w/ S. Ivanov ’12,”vertical Bochner formula”)

For any n and smooth f we have − △(ξf)2 = 2|∇(ξf)|2 − 2df(ξ) · ξ(△f) + 4df(ξ) · g(A, ∇2f) − 4df(ξ)(∇∗A)(∇f).

Lemma (w/ S. Ivanov ’14)

Suppose n = 1, the Lichnerowicz’ condition holds and △f = 2f. We have A(∇f, ∇f) = 0 and (*), i.e., ∇2f(Y, ξ) = df(JY) + 2A(Y, ∇f). Henceforth, if n = 1 we assume the Lichnerowicz’ condition.

CR Obata cont’d (some remarks)

In any dimension:

◮ the Lichnerowicz’ condition implies point-wise

A(∇f, J∇f) ≤ 0;

◮ ∇∗A = 0 ⇒

• M A(∇f, J∇f) Volθ = 0 due to
• M

A(∇f, J∇f) Volθ =

• M

f (∇∗A)(J∇f) Volθ −

• M

g(∇2f, A) Volθ;

◮ hence, (∇∗A)(X) = 0 ⇒ A∇f = 0. Use

∇∗ df(ξ)AJ∇f

• = −A(J∇f, ∇f) + 2|A∇f|2 + df(ξ)(∇∗A)(∇f),

to get

• M

2|A∇f|2 =

• M

A(J∇f, ∇f) = 0.

A = 0 if n = 1

Recall: (i) For k > 0, g(∇f, ∇|A|k) = 0. (ii) |∇f|2|A| = − √ 2A(∇f, J∇f) from Lichnerowicz’ condition. Use an integration by parts argument similar to Li, S.-Y. & Wang, X. ’13 for the case n > 1. For △f = 2f,

◮ using (i), I ≡

• M |A|3f 2(k+1) Volθ= − 1

2

• M |A|3f 2k+1△f Volθ

= 2k+1

2

• M |A|3f 2k|∇f|2 Volθ ≡ 2k+1

2

D.

◮ using (ii),

√ 2(2k + 1)D = −

• M |A|2f 2k+1(∇∗A)(J∇f) Volθ

≤ ||∇∗A||

• M |A|2f 2k+1|∇f| Volθ

≤ ||∇∗A||

a

• M f k+1 f k|∇f| |A|3 Volθ, (suppose |A| ≥ a > 0 so

|A|2 ≤ 1

a|A|3). ◮ H¨

• lder’s inequality

√ 2(2k + 1)D ≤ ||∇∗A||

a

• M |A|3f 2(k+1) Volθ

1/2

M |A|3f 2k|∇f|2 Volθ

1/2 = ||∇∗A||

a

• 2k+1

2

D 1/2 D1/2 = ||∇∗A||

a

• 2k+1

2

1/2 D,

Elliptic equation

Proposition (w/ S. Ivanov ’12)

a) If n > 1, △hf = △f − ∇2f(ξ, ξ) = (2n + 1)f − 1

n(∇∗A)(J∇f).

b) If n = 1, △hf =

• 2 + S−2

6

• f −

1 12g(∇f, ∇S) + 1 3(∇∗A)(J∇f)

In particular, f cannot be a local constant, for f ≡ const.

Sasakain case

In the Sasakian case, Chang, S.-C., & Chiu, H.-L. ’09:

◮ show that the associated Riemannian metric satisfies the Lichnerowicz

condition and achieves equality;

◮ ⇒ isometric to the sphere; ◮ M is of constant Tanaka-Webster curvature.