On the rigidity of Riemannian–Penrose inequality for
asymptotically ﬂat 3-manifolds with corners
· Wenlong Wang
· Haobin Yu
Received: 9 November 2017 / Accepted: 1 April 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this paper we prove a rigidity result for the equality case of the Penrose inequality
on 3-dimensional asymptotically ﬂat manifolds with nonnegative scalar curvature and cor-
ners. Our result also has deep connections with the equality cases of Theorem 1 in Miao
(Commun Math Phys 292(1):271–284, 2009) and Theorem 1.1 in Lu and Miao (Minimal
hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curva-
ture, arXiv:1703.08164v2, 2017).
Keywords Penrose inequality · Asymptotically ﬂat manifold with corner · Stable CMC
Mathematics Subject Classiﬁcation Primary 53C20; Secondary 83C99
In this paper, we are interested in what happens when the equality holds in the Riemannian
Penrose inequality on asymptotically ﬂat manifolds with corners (see Theorem 1.1 below).
Y. Shi and H. Yu: Research partially supported by NSFC 11671015 and NSFC 11731001.
W. Wang: Research partially supported by National Postdoctoral Program for Innovative Talents of China
201700007 and NSFC 11701326.
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking
University, Beijing 100871, People’s Republic of China
Department of Mathematics, Hangzhou Normal University, Hangzhou 310012,
People’s Republic of China