Rigidity of compact quasi-Einstein manifolds with boundary

Abstract

In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We show that a $3$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{3}_{+}$, or the cylinder $I\times\mathbb{S}^2$ with the product metric. For dimension $n=4,$ we prove that a $4$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{4}_{+},$ or the cylinder $I\times\mathbb{S}^3$ with the product metric, or the product space $\mathbb{S}^{2}_{+}\times\mathbb{S}^2$ with the product metric. Other related results for arbitrary dimensions are also discussed.