Isometric embeddings of 2-spheres into Schwarzschild manifolds

Abstract

Let g be a Riemannian metric on the 2-sphere S ². Results on isometric embeddings of (S ², 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 (S ², g) into conformally flat spaces. We prove that if g is close to the standard metric on S ², then (S ², 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.