Remarks on compact quasi-Einstein manifolds with boundary

2021/05/22 by Rafael Diógenes, Diógenes, Rafael, Tiago Gadelha +3 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.2105.10829

To appear in Proc. Amer. Math. Soc

arxiv created 2021/05/22 · arxiv updated 2021/05/25

Abstract

In this paper, we prove that a compact quasi-Einstein manifold (Mⁿ, 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 \BbbSⁿ₊, or g=dt²+ψ²(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.

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