Remarks on compact quasi-Einstein manifolds with boundary

Abstract

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,g,u)$ of dimension $n\ge 4$ with boundary $\partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere ${\mathbb{S}}_{+}^n,$ or $g=d{t}^2+{\psi }^2(t)g_L$ 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.

To appear in Proc. Amer. Math. Soc