Desigualdades geométricas para variedades quasi-Einstein

Abstract

In this work, we establish geometric inequalities for m-quasi-Einstein manifolds, inspired by classical results such as the isoperimetric, Penrose, and Heintze–Karcher inequalities. In particular, we derive estimates for the area of the boundary of compact m-quasi-Einstein manifolds in terms of the first eigenvalue of the Laplace and Jacobi operators, improving previous result sin [32]. Using the generalized Reilly formula obtained by Qiu and Xia [73], we prove an isoperimetric-type inequality for compact m-quasi-Einstein manifolds with boundary, constant scalar curvature, and m > 1. As an application of the boundary area estimate in terms of the first eigenvalue of the Jacobi operator, we establish a Penrose-type inequality for the Hawking mass of the boundary of a three-dimensional compact m-quasi-Einstein manifold. Finally, by applying a generalization of Reilly’s formula due to Li and Xia [51], we obtain a Heintze–Karcher-type inequality valid for compact domains with connected boundary contained in the interior of an m-quasi-Einstein manifold.