Curvature estimates for immersed hypersurfaces in Riemannian manifolds
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.
Abstract In this note, we obtain some results for the Riccati differential equations u′=A(z)+u2 with nonentire meromorphic functions A(z). Some examples are given to illustrate our some results are sharp.