Schwarz type lemmas for pseudo-Hermitian manifolds

TL;DR

This paper establishes Schwarz type lemmas for generalized holomorphic maps, including CR and transversally holomorphic maps, between pseudo-Hermitian manifolds, using Bochner formulas and comparison theorems.

Contribution

It introduces new Schwarz lemmas for pseudo-Hermitian manifolds and extends classical results to CR and transversally holomorphic maps using pseudo-Hermitian analysis.

Findings

Schwarz type results for CR and transversally holomorphic maps

Comparison theorems in pseudo-Hermitian geometry

Discussion on CR hyperbolicity of pseudo-Hermitian manifolds

Abstract

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the \emph{CR} maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the \emph{CR} hyperbolicity problem for pseudo-Hermitian manifolds.