The Einstein-scalar field constraints on asymptotically Euclidean manifolds

TL;DR

This paper develops a method to solve Einstein-scalar field constraint equations on asymptotically Euclidean manifolds, addressing the complexities introduced by scalar fields and allowing for high-dimensional, low-regularity initial data.

Contribution

It extends the conformal method to handle scalar fields in Einstein's equations, providing constructive solutions for arbitrary dimensions and low regularity data.

Findings

Successfully solves Einstein-scalar field constraints using the conformal method.

Handles additional complexity from scalar fields with three extra terms in the equations.

Applicable to high-dimensional and low-regularity initial data scenarios.

Abstract

We use the conformal method to obtain solutions of the Einstein-scalar field gravitational constraint equations. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. Our proofs are constructive and allow for arbitrary dimension (>2) as well as low regularity initial data.