An extension procedure for the constraint equations

TL;DR

This paper introduces a new method to extend solutions of the maximal constraint equations in general relativity from a bounded domain to the entire space, ensuring asymptotic flatness and preserving regularity.

Contribution

It develops a novel approach combining divergence equation solutions and a geometric conformal method, using the implicit function theorem and spherical harmonics expansion.

Findings

Successfully extends solutions from the unit ball to  space

Ensures solutions are asymptotically flat and maintain regularity

Provides a convergent iterative scheme for the extension process

Abstract

Let $( \bar g, \bar k)$ be a solution to the maximal constraint equations of general relativity on the unit ball $B_1$ of $\mathbb{R}^3$. We prove that if $(\bar g,\bar k)$ is sufficiently close to the initial data for Minkowski space, then there exists an asymptotically flat solution $(g,k)$ on $\mathbb{R}^3$ that extends $(\bar g, \bar k)$. Moreover, $(g,k)$ is bounded by $(\bar g, \bar k)$ and has the same regularity. Our proof uses a new method of solving the prescribed divergence equation for a tracefree symmetric $2$-tensor, and a geometric variant of the conformal method to solve the prescribed scalar curvature equation for a metric. Both methods are based on the implicit function theorem and an expansion of tensors based on spherical harmonics. They are combined to define an iterative scheme that is shown to converge to a global solution $(g,k)$ of the maximal constraint equations which extends $(\bar g,\bar k)$.