On the Obata theorem for the weighted Kohn Laplacian in a closed 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 eigenvalue estimates for the weighted Kohn Laplacian.

Contribution

It introduces a weighted CR Reilly's formula and proves the CR Obata theorem in the context of weighted Sasakian manifolds, generalizing previous results.

Findings

Derived a weighted CR Reilly's formula

Established a first eigenvalue estimate for the weighted Kohn Laplacian

Proved the CR Obata theorem in weighted Sasakian manifolds

Abstract

In this paper, we generalize the CR Obata theorem for the Kohn Laplacian to a closed strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the weighted Kohn Laplacian and obtain the corresponding first eigenvalue estimate. With its application, we obtain the CR Obata theorem in a closed weighted Sasakian manifold.