RESEARCH ARTICLE

Characterizations of umbilic hypersurfaces in warped product manifolds

  • Shanze GAO , 1 ,
  • Hui MA 2
Expand
  • 1. School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, China
  • 2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received date: 10 Nov 2020

Accepted date: 15 Apr 2021

Published date: 15 Jun 2021

Copyright

2021 Higher Education Press

Abstract

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.

Cite this article

Shanze GAO , Hui MA . Characterizations of umbilic hypersurfaces in warped product manifolds[J]. Frontiers of Mathematics in China, 2021 , 16(3) : 689 -703 . DOI: 10.1007/s11464-021-0938-1

1
AlÍas L J, Impera D, Rigoli M. Hypersurfaces of constant higher order mean curvature in warped products. Trans Amer Math Soc, 2013, 365(2): 591–621

DOI

2
Barbosa J L, do Carmo M. Stability of hypersurfaces with constant mean curvature. Math Z, 1984, 185(3): 339–353

DOI

3
Barbosa J L, do Carmo M, Eschenburg J. Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math Z, 1988, 197(1): 123–138

DOI

4
Brendle S. Constant mean curvature surfaces in warped product manifolds. Publ Math Inst Hautes Études Sci, 2013, 117: 247–269

DOI

5
Brendle S, Eichmair M. Isoperimetric and Weingarten surfaces in the Schwarzschild manifold. J Differential Geom, 2013, 94(3): 387–407

DOI

6
Kwong K K, Lee H, Pyo J. Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces. Math Res Lett, 2018, 25(2): 597–616

DOI

7
Li H Z, Wei Y, Xiong C W. A note on Weingarten hypersurfaces in the warped product manifold. Internat J Math, 2014, 25(14): 1450121 (13 pp)

DOI

8
Montiel S. Stable constant mean curvature hypersurfaces in some Riemannian manifolds. Comment Math Helv, 1998, 73(4): 584–602

DOI

9
Montiel S. Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds. Indiana Univ Math J, 1999, 48(2): 711–748

DOI

10
O'Neill B. Semi-Riemannian Geometry. Pure Appl Math, Vol 103. New York: Academic Press, 1983

11
Veeravalli A R. Stability of compact constant mean curvature hypersurfaces in a wideclass of Riemannian manifolds. Geom Dedicata, 2012, 159: 1–9

DOI

12
Wu J, Xia C. On rigidity of hypersurfaces with constant curvature functions in warpedproduct manifolds. Ann Global Anal Geom, 2014, 46(1): 1–22

DOI

Outlines

/