A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature

TL;DR

This paper establishes a new sufficient condition for solving the vacuum Einstein constraint equations on compact manifolds using the conformal method, allowing freely specified mean curvature without restrictions.

Contribution

It introduces a condition based on a global supersolution that enables the construction of vacuum solutions with freely specified mean curvature, extending previous results to the vacuum case.

Findings

Provides a new existence theorem for vacuum solutions with arbitrary mean curvature.

Shows vacuum solutions can be obtained as limits of non-vacuum solutions.

Simplifies hypotheses for near-CMC solutions in the conformal method.

Abstract

We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a so-called global supersolution but does not require a global subsolution. As a consequence, we construct a class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, extending a recent result of Holst, Nagy, and Tsogtgerel [HNT07] which constructed similar solutions in the presence of matter. We give a second proof of this result showing that vacuum solutions can be obtained as a limit of [HNT07] non-vacuum solutions. Our principal existence theorem is of independent interest in the near-CMC case, where it simplifies previously known hypotheses required for existence.