Mathematical Research Letters

Volume 25 (2018)

Number 2

Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

Pages: 597 – 616



Kwok-Kun Kwong (Department of Mathematics, National Cheng Kung University, Tainan City, Taiwan)

Hojoo Lee (Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, Korea; and Department of Mathematical Sciences, Seoul National University, Seoul, Korea)

Juncheol Pyo (Department of Mathematics, Pusan National University, Busan, Korea)


We use the weighted Hsiung–Minkowski integral formulas and Brendle’s inequality to show new rigidity results. We prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner–Nordström spaces, where the Alexandrov reflection principle is not available. We also prove that, in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow of codimension one are round hyperspheres.


embedded hypersurfaces, higher order mean curvatures, integral formulas

2010 Mathematics Subject Classification

53C42, 53C44

Part of our work was done while the three authors were visiting the National Center for Theoretical Sciences in Taipei, Taiwan in November 2015. We would like to thank the NCTS for their support and warm hospitality during our visit. We would also like to thank Ye-Kai Wang, Yong Wei and Chao Xia for their useful comments and suggestions. The research of K.-K. Kwong is partially supported by Ministry of Science and Technology in Taiwan under grant MOST103-2115-M-006-016-MY3. J. Pyo is partially supported by the National Research Foundation of Korea (NRF-2015R1C1A1A02036514).

Received 20 September 2016

Accepted 22 January 2017

Published 5 July 2018