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 = λ f . i) λ ≥ n . ii) If ∇ 2 f = − fh , f �≡ 0, then ( M , h ) is isometric with S n ( 1 ) . Notes: 1. Bonnet-Myers: M is compact, diam ( M ) ≤ π and π 1 ( M ) is finite. 2. Lichnerowicz: λ ≥ n , Bochner-Weitzenb¨ ock formula. 3. Obata: if there is λ = n , then ( M , h ) is isometric with S n ( 1 ) . 4. S.Y. Cheng ’75 (improved Toponogov): diam ( M ) = π iff M is isometric to S n ( 1 ) .

Lichnerowicz’ estimate Bochner’s identity ( △ ≥ 0 ) : − 1 2 △|∇ f | 2 = |∇ df | 2 − h ( ∇ ( △ f ) , ∇ f ) + Ric ( ∇ f , ∇ f ) . Therefore M | ( ∇ df ) 0 | 2 + 1 n ( △ f ) 2 − h ( ∇ ( △ f ) , ∇ f ) + Ric ( ∇ f , ∇ f ) dvol . � 0 = For △ f = λ f we get � | ( ∇ df ) 0 | 2 + 1 n λ |∇ f | 2 − λ |∇ f | 2 + Ric ( ∇ f , ∇ f ) dvol 0 = M � � Ric ( ∇ f , ∇ f ) − n − 1 | ( ∇ df ) 0 | 2 dvol + λ |∇ f | 2 dvol = n M M � | ( ∇ df ) 0 | 2 dvol + n − 1 � ( n − λ ) |∇ f | 2 dvol ≥ n M M 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: ◮ ∇ Ric 0 ( ∇ f , X , Y ) = 2 fRic 0 ( X , Y ) + trace term, hence L ∇ f | Ric 0 | 2 k = 4 kf | Ric 0 | 2 k . M | Ric 0 | 2 k f 2 dvol = 1 ◮ � � M h ( ∇| Ric 0 | 2 k f , ∇ f ) dvol n M | Ric 0 | 2 k |∇ f | 2 dvol + 4 k M | Ric 0 | 2 k f 2 dvol . = 1 � � n n M | Ric 0 | 2 k f 2 dvol = M | Ric 0 | 2 k |∇ f | 2 dvol , hence ◮ ( n − 4 k ) � � choosing k > n / 4 it follows Ric 0 = 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 S ETTING

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, J 2 = − id H , 2 g ( X , Y ) def = d θ ( 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 ξ � d θ = 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 ) = ∇ A B − ∇ B A − [ A , B ] satisfies: T ( X , Y ) = 2 ω ( 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 + η 2 ”. Let { ǫ a } 2 n 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 ( ǫ a , A , B , ǫ a ) def = � 2 n a = 1 R ( ǫ a , A , B , ǫ a ) ; scalar curvature S = Ric ( ǫ a , ǫ a ) ; 3. Ricci form: ρ ( A , B ) = 1 2 R ( A , B , ǫ a , J ǫ a ) . Type decomposition of the Ricci tensor: Ric ( X , Y ) = ρ ( JX , Y ) + 2 ( n − 1 ) A ( JX , Y ) . Ricci identity example : ∇ 3 f ( X , Y , ξ ) − ∇ 3 f ( ξ, X , Y ) = ∇ 2 f ( AX , Y ) + ∇ 2 f ( X , AY ) + ( ∇ X A )( Y , ∇ f ) + ( ∇ Y A )( X , ∇ f ) − ( ∇ ∇ f ) A ( X , Y ) . Horizontal divergence e.g.’s: sub-Laplacian: △ f = −∇ 2 f ( ǫ a , ǫ a ) ; 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 2 n + 1 , s.t., for some k 0 = const > 0 we have the Lichnerowicz-type bound Ric ( X , X ) + 4 A ( X , JX ) ≥ k 0 g ( X , X ) , X ∈ H . If n > 1 , then any eigenvalue λ of the sub-Laplacian satisfies n λ ≥ n + 1 k 0 . 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) n If n = 1 the estimate λ ≥ n + 1 k 0 holds assuming in addition that � the CR-Paneitz operator is non-negative M f · Cf Vol θ ≥ 0 , where Cf is the CR-Paneitz operator, Cf = ∇ 4 f ( e a , e a , e b , e b ) + ∇ 4 f ( e a , Je a , e b , Je b ) − 4 n ∇ ∗ A ( J ∇ f ) − 4 n g ( ∇ 2 f , JA ) . Note: Li, S.-Y., & Luk, H.-S. ’04 for n = 1 with condition. Given a function f we define the one form, P f ( X ) = ∇ 3 f ( X , e b , e b ) + ∇ 3 f ( JX , e b , Je b ) + 4 nA ( X , J ∇ f ) so we have Cf = −∇ ∗ P .

The CR-Paneitz operator The divergence formula turns the non-negativity condition ” C ≥ 0” into � � f · Cf Vol θ = − P f ( ∇ f ) Vol θ ≥ 0 . M M ◮ 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 θ = φ 2 θ then ˆ ˆ C = φ − 4 C .

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 C N 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 )+ 2 A ( J ∇ f , ∇ f ) + 4 ∇ df ( ξ, J ∇ f ) . The last term can be related to the traces of ∇ 2 f : � � 1 2 ng ( ∇ 2 f , ω ) 2 + A ( J ∇ f , ∇ f ) Vol θ ∇ 2 f ( ξ, J ∇ f ) Vol θ = − M M and also using the Paneitz operator − 1 2 n ( △ f ) 2 + A ( J ∇ f , ∇ f ) − 1 � � ∇ 2 f ( ξ, J ∇ f ) Vol θ = 2 nP f ( ∇ f ) Vol θ . M M

”key” from the CR Bochner identity Integrating the CR Bochner idenity (for arbitrary function f ) and M ∇ 2 f ( ξ, J ∇ f ) Vol θ term we � using the last two formulas for the find � Ric ( ∇ f , ∇ f ) + 4 A ( J ∇ f , ∇ f ) − n + 1 ( △ f ) 2 Vol θ 0 = n M � 2 − 1 2 n ( △ f ) 2 − 1 � � 2 ng ( ∇ 2 f , ω ) 2 Vol θ � ( ∇ 2 f ) + � � � M � − 3 � � + 2 nP ( ∇ f ) Vol θ . M � � 1 1 Notice that 2 n g , 2 n ω is an orthonormal set in the ( 1 , 1 ) √ √ space with non-zero traces, so − 1 2 n ( △ f ) 2 − 1 2 def 2 � � � � � ( ∇ 2 f ) [ 0 ] � ( ∇ 2 f ) 2 ng ( ∇ 2 f , ω ) 2 . = � � � � � �

Assuming △ f = λ f and the ”Ricci” bound we obtain the inequality: � � k 0 − n + 1 � � � 2 Vol θ |∇ f | 2 Vol θ + � ( ∇ 2 f ) [ 0 ] � � 0 ≥ λ n M M − 3 � P f ( ∇ f ) Vol θ , 2 n M n which implies λ ≥ n + 1 k 0 with equality holding iff ∇ 2 f = 1 2 n ( △ f ) · g + 1 2 ng ( ∇ 2 f , ω ) · ω � and M P f ( ∇ f ) Vol θ = 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 = 2 n + 1 , is a compact spcph manifold which satisfies the Lichnerowicz-type bound. If n ≥ 2 , n then λ = n + 1 k 0 is an eigenvalue iff up-to a scaling ( M , J , θ ) is the standard pseudo-Hermitian CR structure on the unit sphere in C n + 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, A αβ, ¯ β = 0 and A αβ, γ ¯ γ = 0; (ii) for n = 1, A 11 , ¯ 1 = 0 and P 1 f = 0. ◮ w/ S. Ivanov ’12 - assuming ∇ ∗ A = 0 and C ≥ 0 when n = 1.

Q UATERNIONIC C ONTACT S ETTING