An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence free torsion

TL;DR

This paper establishes a characterization of the standard Sasakian sphere as the unique compact pseudohermitian manifold with divergence-free torsion that attains the minimal first eigenvalue of the sub-Laplacian, extending Obata's classical result.

Contribution

It proves a CR Obata type theorem linking the first eigenvalue of the sub-Laplacian to the geometry of pseudohermitian manifolds with divergence-free torsion, including a new version involving traceless horizontal Hessian.

Findings

Characterization of the standard Sasakian sphere via eigenvalue conditions.

Extension of Obata's theorem to CR manifolds with divergence-free torsion.

Existence of functions with traceless horizontal Hessian on complete pseudohermitian manifolds.

Abstract

We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart Sasakian unit sphere. We also give a version of this theorem using the existence of a function with traceless horizontal Hessian on a complete, with respect to Webster's metric, pseudohermitian manifold.