Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds

TL;DR

This paper extends Liouville theorems for semilinear differential inequalities to sub-Riemannian manifolds with nonnegative curvature, including Sasakian manifolds, and provides bounds for lifespan in related inequalities.

Contribution

It generalizes Liouville theorems to a broad class of sub-Riemannian manifolds using new cut-off functions and curvature conditions.

Findings

Liouville theorems hold on sub-Riemannian manifolds with nonnegative curvature

Constructed new classes of cut-off functions for analysis

Provided upper bounds for lifespan in parabolic and hyperbolic inequalities

Abstract

In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.