CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions Shu-Cheng Chang Department of Mathematics and TIMS, NTU (joint work with Ting-Jung Kuo and Jingzhu Tie) Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 1 / 55
Overview Introduction Gradient estimate in Riemannian case 1 Main results 2 Sub-gradient estimate for positive pseudoharmonic functions 1 CR analogue of Liouville-type theorem for positive 2 pseudoharmonic functions The CR Bochner-Type Estimate The Proofs The CR sub-Laplacian comparison property Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 2 / 55
Gradient estimate in Riemannian manifold S.-Y. Cheng and S.-T. Yau derived a well known gradient estimate for positive harmonic functions in a complete noncompact Riemannian manifold. Theorem 2.1 Let M be a complete noncompact Riemannian m-manifold with Ricci curvature bounded from below by � K ( K � 0 ) . If u ( x ) is a positive harmonic function on M , then there exists a positive constant C = C ( m ) such that 2 p jr u j K + 1 � C ( R ) (2.1) u on the ball B ( R ) . As a consequence, the Liouville theorem holds for complete noncompact Riemannian m -manifolds with Ric � 0. Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 3 / 55
CR Yau’s gradient estimates Yau’s Geometric Analysis : Bochner formula 1 Laplacian comparison 2 maximum principle 3 CR Geometric and Analytic aspects : Riemannian Ricci curvature $ pseudohermitian 1 Ricci curvature tensor ( R ij ) and pseudohermitian torsion ( A 11 ) 2 Problem 2.2 Sub-Laplacian ∆ b is degenerated along the missing dirction T by comparing the Riemannian Laplacian ∆ . Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 4 / 55
CR Yau’s gradient estimates By modifying the arguments of Yau, we are able to derive CR version of Yau’s gradient estimate on ( M , J , θ ) with the CR sub-Laplacian comparison property. The same argument can be used to prove CR Li-Yau gradient estimate for positive solutions of the CR heat equation (Chang-Tie-Wu). Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 5 / 55
CR Yau’s gradient estimates De…nition 2.3 Let ( M , J , θ ) be a complete noncompact pseudohermitian ( 2 n + 1 ) -manifold with ( 2 Ric � ( n � 2 ) Tor ) ( Z , Z ) � � 2 k j Z j 2 (2.2) for all Z 2 T 1 , 0 , and k is an nonnegative constant. We say that ( M , J , θ ) satis…es the CR sub-Laplacian comparison property if there exists a positive constant C 0 = C 0 ( k , n ) such that p ∆ b r � C 0 ( 1 r + k ) (2.3) in the sense of distributions. Here r ( x ) is the Carnot-Carathéodory distance from a …xed point x 0 2 M . Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 6 / 55
CR Yau’s gradient estimates It is still not clear that whether the CR sub-Laplacian comparison property holds in a complete noncompact pseudohermitian ( 2 n + 1 ) -manifold ( M , J , θ ) . However, it can be shown that the CR sub-Laplace comparison property holds in the standard Heisenberg ( 2 n + 1 ) -manifold ( H n , J , θ ) . Proposition 2.4 Let ( H n , J , θ ) be a standard Heisenberg ( 2 n + 1 ) -manifold. Then there exists a constant C H n > 0 1 ∆ b r H n � C H n 1 r H n . (2.4) Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 7 / 55
Weak sub-gradient estimate Theorem 3.1 Let ( M , J , θ ) be a complete noncompact pseudohermitian ( 2 n + 1 ) -manifold with ( 2 Ric � ( n � 2 ) Tor ) ( Z , Z ) � � 2 k j Z j 2 �� � � �� � k 1 � � A αβ � , � A αβ , ¯ max α for all Z 2 T 1 , 0 and k � 0 , k 1 > 0 . Assume that ( M , J , θ ) satis…es the CR sub-Laplacian comparison property. If u ( x ) is a positive pseudoharmonic function on M (i.e. ∆ b u = 0 ). Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 8 / 55
Weak sub-gradient estimate Then there exists a small constant ˜ b = ˜ b ( n , k , k 1 ) > 0 and C 4 = C 4 ( k , k 1 , k 2 ) such that for any 0 < b � ˜ b , � � jr b u j 2 + bu 2 u 2 < ( n + 5 ) 2 k + n ( 1 + b ) k 1 + 2 b + C 4 0 (3.1) u 2 5 R on the ball B ( R ) of a large enough radius R which depends only on b . Here u 0 = Tu , and T is the Reeb vector …eld on M . Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 9 / 55
Sub-gradient estimate In order to derive Liouville-type theorem for positive pseudoharmonic functions (i.e. ∆ b u = 0), we need to show a stronger sub-gradient estimate To show the stronger sub-gradient estimate, we have to put the condition [ ∆ b , T ] u = 0 . where T is the Reeb vector …eld. De…nition 3.2 ([GL]) Let ( M , J , θ ) be a pseudohermitian ( 2 n + 1 ) -manifold. We de…ne the purely holomorphic second-order operator Q by n ∑ Qu = 2 i ( A ¯ β u β ) , α . α ¯ α , β = 1 Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 10 / 55
Sub-gradient estimate Theorem 3.3 Let ( M , J , θ ) be a complete noncompact pseudohermitian ( 2 n + 1 ) -manifold with ( 2 Ric � ( n � 2 ) Tor ) ( Z , Z ) � � 2 k j Z j 2 for all Z 2 T 1 , 0 , and k � 0 . Assume that ( M , J , θ ) satis…es the CR sub-Laplacian comparison property. If u ( x ) is a positive pseudoharmonic function with [ ∆ b , T ] u = 0 (3.2) on M. Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 11 / 55
Sub-gradient estimate Then for each constant b > 0, there exists a positive constant C 2 = C 2 ( k ) such that � � jr b u j 2 u 2 < ( n + 5 + 2 bk ) 2 + bu 2 k + 2 b + C 2 0 (3.3) u 2 ( 5 + 2 bk ) R on the ball B ( R ) of a large enough radius R which depends only on b , k . Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 12 / 55
CR Liouville theorem As a consequence, let R ! ∞ and then b ! ∞ with k = 0 in (3.3) , we have the following CR Liouville-type theorem. Corollary 3.4 Let ( M , J , θ ) be a complete noncompact pseudohermitian ( 2 n + 1 ) -manifold with ( 2 Ric � ( n � 2 ) Tor ) ( Z , Z ) � 0 for all Z 2 T 1 , 0 . Assume that ( M , J , θ ) satis…es the CR sub-Laplacian comparison property. If u ( x ) is a positive pseudoharmonic function with [ ∆ b , T ] u = 0 . Then u ( x ) is constant. Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 13 / 55
Liouville-type theorem for Heisenberg manifold Fact 3.5 It is shown that n ∑ [ ∆ b , T ] u = 4 Im [ i ( A ¯ β u β ) , α ] . (3.4) α ¯ α , β = 1 If ( M , J , θ ) is a complete noncompact pseudohermitian ( 2 n + 1 ) -manifold with vanishing torsion. Then [ ∆ b , T ] u = 0 . Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 14 / 55
CR Liouville-type theorem Fact 3.6 On ( H n , J , θ ) , we have R α ¯ β = 0 , and A αβ = 0 . Moreover, [ ∆ b , T ] = 0 holds on ( H n , J , θ ) . Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 15 / 55
CR Liouville-type theorem By applying Corollary 3.4 and Proposition 2.4, we have the following CR Liouville-type theorem for a positive pseudoharmonic function u on ( H n , J , θ ) Corollary 3.7 There does not exist any positive nonconstant pseudoharmonic function in a standard Heisenberg ( 2 n + 1 ) -manifold ( H n , J , θ ) . Koranyi and Stanton proved the Liouville theorem in ( H n , J , θ ) by a di¤erent method via heat kernel. Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 16 / 55
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