Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity

TL;DR

This paper introduces a numerical method using embedding flow and spectral techniques to find isometric embeddings of 2-spheres into Euclidean space, with applications in numerical relativity.

Contribution

The paper develops a spectral method-based embedding flow approach for solving Weyl's embedding problem, extending its application to numerical relativity.

Findings

Successfully computes isometric embeddings of 2-spheres

Demonstrates applicability to quasi-local mass and momentum measures

Provides a framework for coarse-graining in cosmological models

Abstract

We present a numerical method for solving Weyl's embedding problem which consists of finding a global isometric embedding of a positively curved and positive-definite spherical 2-metric into the Euclidean three space. The method is based on a construction introduced by Weingarten and was used in Nirenberg's proof of Weyl's conjecture. The target embedding results as the endpoint of an embedding flow in R^3 beginning at the unit sphere's embedding. We employ spectral methods to handle functions on the surface and to solve various (non)-linear elliptic PDEs. Possible applications in 3+1 numerical relativity range from quasi-local mass and momentum measures to coarse-graining in inhomogeneous cosmological models.