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The goal of this article is to study compact quasi-Einstein manifolds with boundary. We provide boundary estimates for compact quasi-Einstein manifolds similar to previous results obtained for static and 𝑉-static spaces. In addition, we show that compact quasi-Einstein manifolds with connected boundary and satisfying a suitable pinching condition must be isometric, up to scaling, to the standard hemisphere 𝕊 𝑛 + .

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.