Remarks on compact quasi-Einstein manifolds with boundary

TL;DR

This paper classifies compact quasi-Einstein manifolds with boundary under certain curvature conditions, showing they are either standard hemispheres or have a warped product structure, and discusses related cases and examples.

Contribution

It provides a new classification of compact quasi-Einstein manifolds with boundary under curvature assumptions, including a new example and analysis for dimension three.

Findings

Manifolds are either isometric to a hemisphere or have a warped product structure.

A classification result under a fourth-order Weyl tensor vanishing condition.

Introduction of a new example illustrating the assumptions.

Abstract

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,\,g,\,u)$ of dimension $n\geq 4$ with boundary $\partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere $\Bbb{S}^n_+,$ or $g=dt^{2}+\psi ^{2}(t)g_{L}$ and $u=u(t),$ where $g_{L}$ is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension $n=3$ is also discussed.