YAUS GRADIENT ESTIMATE AND LIOUVILLE THEOREM FOR POSITIVE - PDF document

YAU’S GRADIENT ESTIMATE AND LIOUVILLE THEOREM FOR POSITIVE PSEUDOHARMONIC FUNCTIONS IN A COMPLETE PSEUDOHERMITIAN MANIFOLD � SHU-CHENG CHANG 1 , � TING-JUNG KUO 2 , AND JINGZHI TIE 3 Abstract. In this paper, we …rst derive the sub-gradient estimate for positive pseudohar- monic functions in a complete pseudohermitian (2 n +1) -manifold ( M; J; � ) which satis…es the CR sub-Laplacian comparison property. It is served as the CR analogue of Yau’s gradient estimate. Secondly, we obtain the CR sub-Laplacian comparison theorem in a class of com- plete noncompact pseudohermitian manifolds. Finally we have shown the natural analogue of Liouville-type theorems for the sub-Laplacian in a standard Heisenberg (2 n +1) -manifold ( H n ; J ; � ) : 1. Introduction In [Y1] and [CY], S.-Y. Cheng and S.-T. Yau derived a well known gradient estimate for positive harmonic functions in a complete noncompact Riemannian manifold. Proposition 1.1. ( [Y1] , [CY] ) 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 p K + 1 jr f ( x ) j 2 � C ( (1.1) R ) on the ball B ( R ) with f ( x ) = ln u ( x ) : As a consequence, the Liouville theorem holds for complete noncompact Riemannian m -manifolds of nonnegative Ricci curvature. 1991 Mathematics Subject Classi…cation. Primary 32V05, 32V20; Secondary 53C56. Key words and phrases. CR Bochner formula, Subgradient estimate, Liouvile theorem, Pseudohermtian Ricci, Pseudohermitian torsion, Heisenberg group, Pseudohermitian manifold. � Research supported in part by the NSC of Taiwan. 1

� SHU-CHENG CHANG 1 , � TING-JUNG KUO 2 , AND JINGZHI TIE 3 2 In this paper, by modifying the arguments of [Y1], [CY] and [CKL], we derive a sub- gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudo- hermitian (2 n + 1) -manifold ( M; J; � ) which is served as the CR version of Yau’s gradient estimate. Then we prove that the CR analogue of Liouville-type theorem holds for positive pseudoharmonic functions as well. We …rst recall some notions as in section 2 : Let ( M; � ) be a (2 n +1) -dimensional, orientable, contact manifold with contact structure �; dim R � = 2 n . A CR structure J compatible with � is an endomorphism J : � ! � such that J 2 = � 1 . We also assume that J satis…es the integrability condition ( see next section). A CR structure J can extend to C � � and decomposes C � � into the direct sum of T 1 ; 0 and T 0 ; 1 which are eigenspaces of J with respect to eigenvalues i and � i , respectively. A pseudohermitian structure compatible with � is a CR structure J compatible with � together with a choice of contact form � and � = ker � . Such a choice determines a unique real vector …eld T transverse to � which is called the characteristic vector …eld of � , such that � ( T ) = 1 and L T � = 0 or d� ( T; � ) = 0 . Let f T; Z � ; Z � � g be a frame of TM � C , where Z � is any local frame of T 1 ; 0 ; Z � � = Z � 2 T 0 ; 1 . We de…ne Ric and Tor on T 1 ; 0 by � � X � Y � (1.2) Ric ( X; Y ) = R � � and Tor ( X; Y ) = i P � � � A �� X � Y � ) : (1.3) � X � � Y �;� ( A � � � Here X = X � Z � , Y = Y � Z � ; R �� � = R �� � is the pseudohermitian curvature tensor, R � � � � � � � is the pseudohermitian Ricci curvature tensor and A �� is the torsion tensor. In Yau’s method for the proof of gradient estimates, one can estimate �( � ( x ) jr f ( x ) j 2 ) for a nonegative cut-o¤ function � ( x ) on B (2 R ) via Bochner formula and Laplacian com- parison. At the end, one has gradient estimate (1.1) by applying the maximum principle

CR YAU’S GRADIENT ESTIMATE AND LIOUVILLE THEOREM 3 to � ( x ) jr f ( x ) j 2 : However in order to derive the CR subgradient estimate, one of di¢cul- ties is to deal with the following CR Bochner formula (Lemma 2.1) which involving a term h J r b '; r b ' 0 i that has no analogue in the Riemannian case. � � � r H � 2 ' 2 � � � b jr b ' j 2 = 2 � � + 2 hr b '; r b � b ' i + (4 Ric � 2 ( n � 2) Tor ) (( r b ' ) C ; ( r b ' ) C ) + 4 h J r b '; r b ' 0 i : � r H � 2 ; � b , r b are the subhessian, sub-Laplacian and sub-gradient respectively. We Here also denote ' 0 = T' . In order to overcome this di¢culty, we introduce a real-valued function F ( x; t; R; b ) : 0 ( x ) to jr b f ( x ) j 2 as M � [0 ; 1] � (0 ; 1 ) � (0 ; 1 ) ! R by adding an extra term t� ( x ) f 2 following � � jr b f ( x ) j 2 + bt� ( x ) f 2 F ( x; t; R; b ) = t 0 ( x ) on the Carnot-Carathéodory ball B (2 R ) with a constant b to be determined. In section 3 , we derive the CR subgradient estimate (1.11) and (1.8) by applying the maximum principle to � ( x ) F ( x; t ) for each …xed t 2 (0 ; 1] if the CR sub-Laplacian comparison property (see De…nition 1.2) holds on ( M; J; � ) . De…nition 1.1. Let ( M; J; � ) be a pseudohermitian (2 n + 1) -manifold. A piecewise smooth curve � : [0 ; 1] ! M is said to be horizontal if � 0 ( t ) 2 � whenever � 0 ( t ) exists. The length of � is then de…ned by Z 1 1 h � 0 ( t ) ; � 0 ( t ) i l ( � ) = L � dt: 2 0 Here h ; i L � is the Levi form as in (2.2). The Carnot-Carathéodory distance between two points p , q 2 M is d c ( p; q ) = inf f l ( � ) j � 2 C p;q g where C p;q is the set of all horizontal curves joining p and q . We say M is complete if it is complete as a metric space. We refer to [S] for some details. By Chow connectivity

� SHU-CHENG CHANG 1 , � TING-JUNG KUO 2 , AND JINGZHI TIE 3 4 theorem [Cho] , there always exists a horizontal curve joining p and q , so the distance is …nite. Furthermore, there is a minimizing geodesic joining p and q so that its length is equal to the distance d c ( p; q ) . De…nition 1.2. Let ( M; J; � ) be a complete noncompact pseudohermitian (2 n +1) -manifold with (2 Ric � ( n � 2) Tor ) ( Z; Z ) � � 2 k j Z j 2 (1.4) for all Z 2 T 1 ; 0 , and k is an nonnegative constant. We say that ( M; J; � ) 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 (1.5) r + k ) in the sense of distributions. Here � b denote sub-Laplacian and r ( x ) is the Carnot-Carathéodory distance from a …xed point x 0 2 M: Let ( M; J; � ) be the standard Heisenberg (2 n + 1) -manifold ( H n ; J ; � ) : We have R � � � = 0 and A �� = 0 : Then the following CR sub-Laplacian comparison property holds on ( H n ; J ; � ) . Proposition 1.2. ( [CTW] ) Let ( H n ; J; � ) 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 (1.6) r H n : Remark 1.1. 1. In the paper [CTW] , it was shown tha the CR sub-Laplacian comparison property (1.5) holds in a complete Heisenberg (2 n + 1) -manifold ( H n ; J ; � ) : Here we will give an another proof by applying the di¤erential inequality for sub-Laplacian of Carnot- Caratheodory distance as in Lemma 4.1.

CR YAU’S GRADIENT ESTIMATE AND LIOUVILLE THEOREM 5 2. We expect that the method will be able to adapt to derive the CR sub-Laplacian compar- ison property in a complete noncompact pseudohermitian manifold other than the standard Heisenberg manifold. In particular, (1.5) holds in an asymptotic Heisenberg manifold as well, we will discuss it elsewhere. We refer to Theorem 4.3 as in section 4 for some details. In order to have an analogue of Liouville-type theorem ( see Corollary 1.4 ) for positive pseudoharmonic functions (i.e. � b u = 0 ) in a complete noncompact pseudohermitian (2 n + 1) -manifold, we need to show the following sub-gradient estimate for positive pseudoharmonic functions u: Theorem 1.3. Let ( M; J; � ) 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 : Furthermore, we assume that ( M; J; � ) satis…es the CR sub- Laplacian comparison property (1.5). If u ( x ) is a positive pseudoharmonic function with (1.7) [� b ; T ] u = 0 on M . 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 (1.8) u 2 (5 + 2 bk ) R on the ball B ( R ) of a large enough radius R which depends only on b , k . As a consequence, let R ! 1 and then b ! 1 with k = 0 in (1.8) ; we have the following CR Liouville-type theorem. Corollary 1.4. Let ( M; J; � ) be a complete noncompact pseudohermitian (2 n +1) -manifold with (2 Ric � ( n � 2) Tor ) ( Z; Z ) � 0