On Weyl's embedding problem in Riemannian manifolds

Abstract

We consider a priori estimates of Weyl's embedding problem of $({\mathbb{S}}^2,g)$ in general $3$-dimensional Riemannian manifold $(N^3,\bar{g})$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang, we obtain an isometric embedding of $({\mathbb{S}}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl's embedding theorem in space form under the condition that $g\in C^2$ with $D^{2}g$ Dini continuous.

To appear in Int. Math. Res. Not. IMRN. https://doi.org/10.1093/imrn/rny109

References in corpus (3)

• Isometric embeddings into the Minkowski space and new quasi-local mass
• Curvature estimates for immersed hypersurfaces in Riemannian manifolds
• Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

Cited by in corpus (3)

• Curvature estimates for immersed hypersurfaces in Riemannian manifolds
• A rigidity theorem for surfaces in Schwarzschild manifold
• Uniqueness of isometric immersions with the same mean curvature