Rigidity of Einstein metrics as critical points of some quadratic curvature functionals on complete manifolds

TL;DR

This paper investigates the rigidity of Einstein metrics on complete manifolds by analyzing quadratic curvature functionals through point-wise and integral inequalities involving curvature tensors.

Contribution

It introduces new rigidity results for Einstein metrics as critical points of quadratic curvature functionals using point-wise and integral inequalities.

Findings

Rigidity results for Einstein metrics via point-wise inequalities

Rigidity results involving integral inequalities and curvature tensors

Conditions involving Sobolev constants for Einstein metric rigidity

Abstract

In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the trace-less Ricci curvature and the Sobolev constant, accordingly.