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-Nordstr\"om manifolds

The Weyl problem in warped product spaces

“…Hence we need to deal with closeness, openness, rigidity and convexity. Recently, the closeness has been obtained in [11]. In the first part of this paper, we establish the openness for strictly convex surface in any 3-dimensional warped product spaces.…”

“…Here the assumption (a), (b) and the constant K 0 is defined by (5.1) in section 5. The above existence theorem is proposed and proved by Guan-Lu [11] and Lu [27].…”

A free boundary isometric embedding problem in the unit ball

“…We note that the isometric embedding problem for more general target manifolds, particularly with warped product metrics, has been studied. We refer to the works of P. Guan and S. Lu, see [9], S. Lu, see [22], as well as C. Li and Z. Wang, see [23].…”

The Weyl problem of isometric immersions revisited

“…also Chang–Xiao [3]. For the existence results of general ambient manifolds other than space forms, we refer to Pogorelov [28] and recent works by Guan–Lu [8] and Li–Wang [20].…”