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 ≥ 4 with boundary ∂M, nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere S n + , or g = dt 2 + ψ 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.