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

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Abstract

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.

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Acknowledgments

Both authors would like to thank Professors Guofang Wang, Yuxin Ge and Dr. Wei Wang for helpful discussions.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei,  230026, People’s Republic of China

    Jie Wu

  2. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, 79104 , Freiburg, Germany

    Jie Wu

  3. Max-Planck-Institut für Mathematik in den Naturwissenschaft, Inselstr. 22, 04103 , Leipzig, Germany

    Chao Xia

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  1. Jie Wu
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  2. Chao Xia
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Correspondence to Chao Xia.

Additional information

J. Wu is partly supported by SFB/TR71 “Geometric partial differential equations” of DFG. C. Xia is supported by funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 267087.

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Wu, J., Xia, C. On rigidity of hypersurfaces with constant curvature functions in warped product manifolds. Ann Glob Anal Geom 46, 1–22 (2014). https://doi.org/10.1007/s10455-013-9405-x

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