Curvature estimates for immersed hypersurfaces in Riemannian manifolds

Pengfei Guan; Siyuan Lu

doi:10.1007/s00222-016-0688-y

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arXiv · 1604.06150

Curvature estimates for immersed hypersurfaces in Riemannian manifolds

Pengfei Guan·Siyuan Lu

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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-Nordstr\"om manifolds.

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Pengfei Guan, Siyuan Lu. 2016-04-21. Curvature estimates for immersed hypersurfaces in Riemannian manifolds. https://doi.org/10.1007/s00222-016-0688-y

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