Rigidity of compact quasi-Einstein manifolds with boundary

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.