Liouville Theorems for holomorphic maps on pseudo-Hermitian manifolds

TL;DR

This paper establishes Liouville theorems for various classes of holomorphic maps between pseudo-Hermitian and almost Hermitian manifolds, demonstrating conditions under which such maps must be constant.

Contribution

It proves new Liouville type results for holomorphic maps involving pseudo-Hermitian and almost Hermitian manifolds, including constantness under curvature conditions.

Findings

Holomorphic maps from pseudo-Hermitian to almost Hermitian manifolds are constant under certain curvature conditions.

Constructs explicit almost CR structures on complex vector bundles over almost CR manifolds.

Shows that maps with specific curvature assumptions are necessarily trivial.

Abstract

We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds, assuming that the domains are compact. For instance, we show that any $(J,J^N)$ holomorphic map from a compact pseudo-Hermitian manifold $M$ with nonnegative (resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian manifold $N$ with negative (resp. nonpositive) holomorphic sectional curvature is constant. We also construct explicit almost CR structures on a complex vector bundle over an almost CR manifold.