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Curvature estimates for immersed hypersurfaces in Riemannian manifolds

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Abstract

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold \((N^{n+1}, \bar{g})\) through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of \(({\mathbb {S}}^2, g)\) in 3-dimensional warped product space \((N^3, \bar{g})\). We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter–Schwarzschild manifolds and the Reissner–Nordström manifolds.

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Acknowledgments

P. Guan would like to thank Mu-Tao Wang for enlightening conversations regarding the isometric embedding problems and the quasi-local masses in general relativity. We would like to thank the anonymous referee for the help in the exposition of the paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke O, Montreal, QC, H3A 0B9, Canada

    Pengfei Guan & Siyuan Lu

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  1. Pengfei Guan
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  2. Siyuan Lu
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Correspondence to Pengfei Guan.

Additional information

Research of P. Guan was supported in part by an NSERC Discovery Grant. Research of S. Lu was supported in part by CSC fellowship and Schulich Graduate fellowship.

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Guan, P., Lu, S. Curvature estimates for immersed hypersurfaces in Riemannian manifolds. Invent. math. 208, 191–215 (2017). https://doi.org/10.1007/s00222-016-0688-y

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  • DOI: https://doi.org/10.1007/s00222-016-0688-y

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