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Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds

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Abstract

Given a compact Riemannian manifold M, we consider a warped product manifold \({\bar{M}} = I \times _h M\), where I is an open interval in \({\mathbb {R}}\). For a positive function \(\psi \) defined on \({\bar{M}}\), we generalize the arguments in Guan et al. (Commun. Pure Appl. Math. 68(8):1287–1325, 2015) and Ren and Wang (On the curvature estimates for Hessian equations, 2016. arXiv:1602.06535), to obtain the curvature estimates for Hessian equations \(\sigma _k(\kappa )=\psi (V,\nu (V))\). We also obtain some existence results for the starshaped compact hypersurface \(\Sigma \) satisfying the above equation with various assumptions.

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Acknowledgements

The last author wish to thank Professor Pengfei Guan for some discussions about constant rank theorems. He also would like to thank Tsinghua University for their support and hospitality during the paper being prepared. The authors would also like to thank the referees for some helpful comments which made this paper more readable.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Daguang Chen & Haizhong Li

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Zhizhang Wang

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  1. Daguang Chen
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  2. Haizhong Li
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  3. Zhizhang Wang
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Correspondence to Zhizhang Wang.

Additional information

Communicated by O. Savin.

The Daguang Chen was supported by NSFC Grant No. 11471180, the Haizhong Li was supported by NSFC Grant No. 11671224, and the Zhizhang Wang was partially supported by NSFC Grants Nos. 11301087, 11671090 and 11771103.

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Chen, D., Li, H. & Wang, Z. Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds. Calc. Var. 57, 42 (2018). https://doi.org/10.1007/s00526-018-1314-1

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