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Tian Chong

Publications and source records attributed to Tian Chong.

10 recordsLinked to original sources
arXiv2025

On critical maps of the horizontal energy functional between Riemannian foliations

In this paper, we consider critical points of the horizontal energy $E_{\HH}(f)$ for a smooth map $f$ between two Riemannian foliations. These critical points are referred to as horizontally harmonic maps. In particular, if the maps are foliated, they become transversally harmonic maps. By utilizing the stress-energy tensor, we establish some monotonicity formulas for horizontally harmonic maps from Euclidean spaces, the quotients $K_{m}$ of Heisenberg groups and also for transversally harmonic maps from Riemannian foliations with appropriate curvature pinching conditions. Finally, we give Jin-type theorems for either horizontally harmonic maps or transversally harmonic maps under some asymptotic conditions at infinity.

math.DG↗
arXiv2022

On stability of subelliptic harmonic maps with potential

In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a subelliptic harmonic map with potential is stable if the target manifold has nonpositive curvature and the Hessian of the potential is nonpositive definite. We also give Leung type results which involve the instability of subelliptic harmonic maps with potential when the target manifold is a sphere of dimension $\geq 3$.

math.DG↗
arXiv2020

Schwarz type lemmas for generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carathéodory pseudodistance on CR manifolds.

math.DG↗
arXiv2019

Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds To Regular Balls

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 pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls of Riemannian manifolds.

math.DG↗
arXiv2016

Liouville Theorems for critical points of the p-Ginzburg-Landau type functional

In this paper, we consider the smooth map from a Riemannian manifold to the standard Euclidean space and the p-Ginzburg-Landau energy. Under suitable curvature conditions on the domain manifold, some Liouville type theorems are established by assuming either growth conditions of the p-Ginzburg-Landau energy or an asymptotic condition at the infinity for the maps. In the end of paper, we obtain the unique constant solution of the constant Dirichlet boundary value problems on starlike domains.

math.DG↗
arXiv2014

Rigidity theorems of complete Kähler-Einstein manifolds and complex space forms

We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kähler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among Kähler manifolds with constant scalar curvature.

math.DG↗
arXiv2014

Unstability of pseudoharmonic maps between pseudo-Hermitian manifolds

In this paper, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian manifolds. When the target manifold is an isometric embedded CR manifold in complex Euclidean space or a pseudo-Hermitian immersed submanifold in Heisenberg group, we give some conditions on Weingarten maps to obtain some unstability of pseudoharmonic maps between these pseudo-Hermitian manifolds.

math.DG↗
arXiv2013

Liouville theorem for Pseudoharmonic maps from Sasakian manifolds

In this paper, we derive a sub-gradient estimate for pseudoharmonic maps from noncompact complete Sasakian manifolds which satisfy CR sub-Laplace comparison property, to simply-connected Riemannian manifolds with nonpositive sectional curvature. As its application, we obtain some Liouville theorems for pseudoharmonic maps. In the Appendix, we modify the method and apply it to harmonic maps from noncompact complete Sasakian manifolds.

math.DG↗