Annals of Global Analysis and Geometry

, Volume 46, Issue 1, pp 1–22

On rigidity of hypersurfaces with constant curvature functions in warped product manifolds



In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as “weighted” mean curvatures, which extend the work (Brendle in Publ Math Inst Hautes Études Sci 117:247–269, 2013; Brendle and Eichmair in J Differ Geom 94(94):387–407, 2013; Montiel in Indiana Univ Math J 48:711–748, 1999) considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss–Bonnet curvatures \(L_k\). To achieve this, we develop some new kind of Newton–Maclaurin type inequalities on \(L_k\) which may have independent interest.


Constant mean curvature Rigidity Warped product manifold Gauss–Bonnet curvature 

Mathematics Subject Classification (2010)

Primary 53C24 Secondary 52A20 53C40 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany
  3. 3.Max-Planck-Institut für Mathematik in den NaturwissenschaftLeipzigGermany

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