Curvature estimates for immersed hypersurfaces in Riemannian manifolds

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.

References in corpus (4)

• Isometric embeddings into the Minkowski space and new quasi-local mass
• The Weyl problem in warped product space
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Cited by in corpus (14)

• Interior C2 regularity of convex solutions to prescribing scalar curvature equations
• On Weyl's embedding problem in Riemannian manifolds
• The global curvature estimate for the n-2 Hessian equation
• Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity
• Starshaped compact hypersurfaces with prescirbed Weingarten curvature in warped product manifolds
• A rigidity theorem for surfaces in Schwarzschild manifold
• The curvature estimates for convex solutions of some fully nonlinear Hessian type equations
• Notes on the curvature estimates for Hessian equations
• On the local isometric embedding of trapped surfaces into three-dimensional Riemannian manifolds
• Counterexamples to the $L^p$-Calderón--Zygmund Estimate on Open Manifolds
• The Weyl problem of isometric immersions revisited
• Recognition of affine-equivalent polyhedra by their natural developments
• Weyl estimates for spacelike hypersurfaces in de Sitter space
• Uniqueness of isometric immersions with the same mean curvature