Horizontal Gradient Estimate of Positive Pseudo-Harmonic Functions on Complete Noncompact Pseudo-Hermitian Manifolds

TL;DR

This paper derives a horizontal gradient estimate for positive solutions of a sub-Laplacian equation on complete noncompact pseudo-Hermitian manifolds, leading to a Liouville theorem for positive pseudo-harmonic functions.

Contribution

It provides a new gradient estimate for positive solutions of the sub-Laplacian on pseudo-Hermitian manifolds, extending Liouville theorems to broader geometric contexts.

Findings

Gradient estimate for positive solutions of Δ_b u = -λ u

Recovers Liouville theorem for positive pseudo-harmonic functions on Sasakian manifolds

Applicable to manifolds with nonnegative pseudo-Hermitian Ricci curvature

Abstract

In this paper, we will give a horizontal gradient estimate of positive solutions of $\Delta_b u = - \lambda u$ on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive pseudo-harmonic functions on Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.