On the Obata Theorem in a weighted Sasakian manifold

TL;DR

This paper extends the CR Obata theorem to weighted Sasakian manifolds by deriving a weighted CR Reilly's formula and establishing a first eigenvalue estimate, leading to a generalized Obata theorem in this setting.

Contribution

The paper introduces a weighted CR Reilly's formula and applies it to prove a generalized CR Obata theorem for compact weighted Sasakian manifolds.

Findings

Derived a weighted CR Reilly's formula for the Witten sub-Laplacian

Established a first eigenvalue estimate in the weighted setting

Proved the CR Obata theorem for compact weighted Sasakian manifolds

Abstract

In this paper, we generalize the CR Obata theorem to a compact strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the Witten sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its applications, we obtain the CR Obata theorem in a compact weighted Sasakian manifold with or without boundary.