Curvature estimates for immersed hypersurfaces in Riemannian manifolds

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold (Nn+1,g¯)$$(N^{n+1}, \bar{g})$$ 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 (S2,g)$$({\mathbb {S}}^2, g)$$ in 3-dimensional warped product space (N3,g¯)$$(N^3, \bar{g})$$. 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.