Liouville Theorems on pseudohermitian manifolds with nonnegative Tanaka-Webster curvature

TL;DR

This paper proves a rigidity theorem for positive solutions to the CR Yamabe equation on noncompact Sasakian manifolds with nonnegative Tanaka-Webster curvature, extending known results to curved settings and characterizing the Heisenberg group uniquely.

Contribution

It extends Jerison-Lee's rigidity result to curved Sasakian manifolds, showing the Heisenberg group is uniquely characterized by positive solutions under nonnegative curvature.

Findings

Heisenberg group is the only complete Sasakian space with nonnegative Tanaka-Webster scalar curvature admitting positive solutions.

Under natural assumptions, the rigidity result extends to higher dimensions.

The work generalizes known flat case results to curved manifolds.

Abstract

In this paper we study positive solutions to the CR Yamabe equation in noncompact $(2n+1)$-dimensional Sasakian manifolds with nonnegative curvature. In particular, we show that the Heisenberg group $\mathbb{H}^1$ is the only (complete) Sasakian space with nonnegative Tanaka-Webster scalar curvature admitting a (nontrivial) positive solution. Moreover, under some natural assumptions, we prove this strong rigidity result in higher dimensions, extending the celebrated Jerison-Lee's result to curved manifolds.