  In this article, we study the geometry of compact quasi-Einstein manifolds
with boundary. We establish sharp boundary estimates for compact quasi-Einstein
manifolds with boundary that improve some previous results. Moreover, we obtain
a characterization theorem for such manifolds in terms of the surface gravity
of the boundary components, which leads to a new sharp geometric inequality. In
addition, we prove a boundary estimate for compact quasi-Einstein manifolds
with (possibly disconnected) boundary in terms of the Brown-York mass.
