Geometric inequalities for quasi-Einstein manifolds

TL;DR

This paper establishes new geometric inequalities for compact quasi-Einstein manifolds with boundary, including boundary estimates, isoperimetric inequalities, and relations involving eigenvalues and Hawking mass, using generalized Reilly's formulas.

Contribution

It introduces novel boundary estimates and inequalities for quasi-Einstein manifolds, extending classical results with new geometric bounds and relations.

Findings

Derived boundary estimates in terms of the first eigenvalue of the Jacobi operator.

Established an isoperimetric type inequality for compact quasi-Einstein manifolds.

Presented a Heintze-Karcher type inequality for compact domains in these manifolds.

Abstract

In this article, we investigate certain geometric inequalities on quasi-Einstein manifolds. We use the generalized Reilly's formulas by Qiu-Xia and Li-Xia to establish new boundary estimates and an isoperimetric type inequality for compact quasi-Einstein manifolds with boundary. Boundary estimates in terms of the first eigenvalue of the Jacobi operator and the Hawking mass are also established. In particular, we present a Heintze-Karcher type inequality for compact domains in quasi-Einstein manifolds.