An Obata-type theorem on a three-dimensional CR manifold

TL;DR

This paper establishes a CR geometric analogue of Obata's theorem, characterizing the standard three-dimensional sphere via the first eigenvalue of the sub-Laplacian on certain pseudohermitian manifolds.

Contribution

It proves a new Obata-type theorem in CR geometry, linking the first eigenvalue of the sub-Laplacian to the standard sphere under specific curvature conditions.

Findings

First eigenvalue characterization of the sphere

Extension of Obata's theorem to CR manifolds

Conditions for equality case in eigenvalue estimate

Abstract

We prove a CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Panietz operator which satisfies a Lichnerowicz type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three dimensional unit sphere.