Curvature estimates for immersed hypersurfaces in Riemannian manifolds

TL;DR

This paper derives mean curvature estimates for immersed hypersurfaces with nonnegative extrinsic scalar curvature in Riemannian manifolds, impacting isometric embedding problems in general relativity and warped product spaces.

Contribution

It introduces a new regularity approach to a degenerate fully nonlinear curvature equation, enabling curvature estimates in general Riemannian manifolds.

Findings

Mean curvature estimate for hypersurfaces with nonnegative extrinsic scalar curvature.

Application to Weyl isometric embedding problem in warped product spaces.

Discussion of isometric embedding in spaces with horizons like Anti-de Sitter-Schwarzschild.

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.