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

TL;DR

This paper develops gradient estimates and existence results for pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to Riemannian balls, extending classical geometric analysis tools to pseudo-Hermitian settings.

Contribution

It introduces a sub-Laplacian estimate for distance functions in pseudo-Hermitian geometry and establishes existence and Liouville theorems for pseudo-harmonic maps.

Findings

Derived a sub-Laplacian comparison theorem in pseudo-Hermitian geometry.

Established a priori gradient estimates for pseudo-harmonic maps.

Proved existence of pseudo-harmonic maps under certain curvature conditions.

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