On Weyl’s Embedding Problem in Riemannian Manifolds

Abstract

We consider a priori estimates of Weyl's embedding problem of (S 2 , g) in general 3-dimensional Riemannian manifold (N 3 ,ḡ). 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 (S 2 , g) in Riemannian manifold. In addition, we reprove Weyl's embedding theorem in space form under the condition that g ∈ C 2 with D 2 g Dini continuous.

Uniqueness of isometric immersions with the same mean curvature

“…If the target space is a general Riemannian 3-manifold N, estimates concerning the existence of isometric embeddings into N have been studied extensively in the literature. We refer readers to the work in [9,19,24,25,30,34,35] and the references therein.…”

Capacity, quasi-local mass, and singular fill-ins

“…It was shown in [20,Theorem 1.3] (see also [19]) that Λ(Σ, γ) < ∞ for finite unions of topological spheres. By convention, Λ(Σ, γ) := −∞ when F (Σ, γ) = ∅; likewise forΛ,F .…”

Total Mean Curvature, Scalar Curvature, and a Variational Analog of Brown–York Mass

“…Remark 1.11. After the first version of this paper was completed, we learned that the assertion Λ (Σ,γ) < ∞ in Theorem 1.3, and its proof, appear independently in a recent work by Lu [8], which establishes a priori estimates for certain isometric embeddings of 2-spheres into general Riemannian 3-manifolds.…”

“…The first named author would like to thank Rick Schoen for his continued guidance and Otis Chodosh for a helpful discussion in the early stages of this work. Both authors would like to thank Pengfei Guan for bringing the work in [8] to our attention, as well as the referees for their helpful comments and their careful reading of the manuscript.…”