Isometric embeddings of 2-spheres into Schwarzschild manifolds

Abstract

Let g be a Riemannian metric on the 2-sphere S2. Results on isometric embeddings of (S2, g) into a fixed model manifold often have implications in quasi-local mass related problems in general relativity. In this paper, motivated by the definitions of the Brown–York and the Wang–Yau mass, we consider isometric embeddings of (S2, g) into conformally flat spaces. We prove that if g is close to the standard metric on S2, then (S2, g) admits an isometric embedding into any spatial Schwarzschild manifold with small mass. We also give a sufficient condition that ensures isometric embeddings of perturbations of a Euclidean convex surface into \({\mathbb{R}^3}\) equipped with a conformally flat metric.