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) over bar )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 (S-2, g) in 3-dimensional warped product space (N-3,(g) over bar) . We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordstrom manifolds.