CR Sub-Laplacian Comparison and Liouville-type Theorem in a Complete Noncompact Sasakian Manifold

TL;DR

This paper establishes a sub-Laplacian comparison theorem and Liouville-type results for positive pseudoharmonic functions on complete noncompact Sasakian manifolds, extending classical geometric analysis results to CR geometry.

Contribution

It introduces the CR sub-Laplacian comparison theorem and derives gradient estimates, providing new tools for analysis on Sasakian manifolds.

Findings

Sub-Laplacian comparison theorem proved for Sasakian manifolds.

Gradient estimates for positive pseudoharmonic functions established.

Liouville-type theorems derived for nonnegative pseudohermitian Ricci curvature cases.

Abstract

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the CR sub-Laplacian comparison property. It is served as the CR analog of Yau's gradient estimate. As a consequence, we have the natural CR analogue of Liouville-type theorems in a complete noncompact Sasakian manifold of nonnegative pseudohermitian Ricci curvature tensors.