The Isometric Embedding Problem in a Null Cone

TL;DR

This paper extends isometric embedding results to arbitrary dimensions and null cones, providing new existence, uniqueness, and asymptotic embedding theorems for metrics on spheres within complex geometric settings.

Contribution

It generalizes the linearized embedding problem to higher dimensions and null cones, and establishes new embedding and foliation results for 2-sphere metrics.

Findings

Extended Li-Wang's linearized embedding results to arbitrary dimensions.

Proved metric perturbations of spheres are isometrically embeddable up to Codazzi solutions.

Established existence and uniqueness of embeddings in null cone settings.

Abstract

In the first part of this paper, we extend the result of Li-Wang on the linearized embedding problem to a compact manifold of arbitrary dimension. Using this, we then show that any metric perturbation of a embedded $n$-sphere is also isometrically embedded up to a solution of the homogenous Codazzi equation, irrespective of the ambient geometry. In the second part we specialize to dimension two, and study these results within an ambient Null Cone. Specifically, given a path of metrics on the 2-sphere and an initial isometric embedding, we develop a small parameter existence and uniqueness theorem for paths of isometric embeddings. In the final part, after imposing asymptotic decay conditions on the Null Cone, we show that any metric on the 2-sphere can be isometrically embedded up to a scaling factor. We then prove the existence of a foliation in a neighborhood of infinity.