Pseudo-harmonic Maps from Complete Noncompact Pseudo-Hermitian Manifolds to Regular Balls
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Pseudo-harmonic Maps from Complete Noncompact Pseudo-Hermitian Manifolds to Regular Balls Tian Chong1 · Yuxin Dong2 · Yibin Ren3
· Wei Zhang4
Received: 24 December 2018 © Mathematica Josephina, Inc. 2019
Abstract In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudoHermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. Keywords Sub-Laplacian comparison theorem · Regular ball · Pseudo-harmonic maps · Horizontal gradient estimate · Liouville theorem · Existence theorem
Y. Dong: Supported by NSFC Grant No. 11771087 and LMNS, Fudan. Y. Ren: Supported by NSFC Grant No. 11801517.
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Yibin Ren [email protected] Tian Chong [email protected] Yuxin Dong [email protected] Wei Zhang [email protected]
1
School of Science, College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, People’s Republic of China
2
School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
3
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, Zhejiang, People’s Republic of China
4
School of Mathematics, South China University of Technology, Guangzhou 510641, People’s Republic of China
123
T. Chong et al.
Mathematics Subject Classification 58E20 · 53C25 · 32V05
1 Introduction Inspired by Eells–Sampson’s theorem, one natural problem is to consider the existence of harmonic maps from complete noncompact Riemannian manifolds. Usually some convexity conditions on the images will lead this existence (cf. [10,17,18]). Based on elliptic theory, some existence theorems have been studied for generalized harmonic maps (cf. [7,20]). The pseudo-harmonic map is an analogue of the harmonic map in pseudo-Hermitian geometry. Let (M, θ ) be a pseudo-Hermitian manifold of real dimension 2m + 1 and (N , h) be a Riemannian manifold. The horizontal energy of a smooth map f : M → N is defined by |db f |2 θ ∧ (dθ )m , (1.1) EH ( f ) = M
where db f is the horizontal part of d f . The pseudo-harmonic map is a critical point of E H . Hence it locally satisfies the following Euler–Lagrange equation
τ Hi ( f ) = b f i +
ijk ( f )db f j , db f k = 0,
(1.2)
j,k
where ijk ’s are Christoffel symbols of Levi-Civita connection in (N , h). Here b denotes the sub-Laplacian which is a subelliptic operator enjoying nice regularity as elliptic operators. By heat flow method, the Eells–Sampson’s type theorem also holds for pseudo-harmonic maps (cf. [5,21]). The Dirichlet problem of pseudo-harmonic
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