Compact gradient Einstein-type manifolds with boundary and constant scalar curvature

TL;DR

This paper proves rigidity results for compact gradient Einstein-type manifolds with boundary and constant scalar curvature, extending previous classification results and including special cases like $(m, ho)$-quasi-Einstein manifolds.

Contribution

It extends classification results to manifolds with boundary under pinching conditions and introduces new rigidity results for specific Einstein-type manifolds.

Findings

Rigidity results for compact gradient Einstein-type manifolds with boundary.

Extension of classification to manifolds with boundary under pinching conditions.

Rigidity results for $(m, ho)$-quasi-Einstein manifolds with boundary.

Abstract

Inspired by the study of $V$-static manifold about classification, in this article, we apply the recent results obtained by Freitas and Gomes (Compact gradient Einstein-type manifolds with boundary, 2022) to prove the rigidity results for compact gradient Einstein-type manifolds with nonempty boundary and constant scalar curvature under some suitable pinching conditions. As a special case of gradient Einstein-type manifold, we also give a rigidity result of $(m,\rho)$-quasi-Einstein manifold with boundary.