Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions

TL;DR

This paper proves the existence of non-constant mean curvature solutions to Einstein's constraint equations on closed manifolds without the near-CMC assumption, extending previous results to all Yamabe classes and weaker regularity conditions.

Contribution

It establishes the first non-CMC existence results without near-CMC restrictions on closed manifolds with positive Yamabe class, using weak background metrics and small matter data.

Findings

Existence of non-CMC solutions without near-CMC assumption.

Results valid for all Yamabe classes.

Extended regularity to maximum allowed by data.

Abstract

We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions for the Hamiltonian constraint, and topological fixed-point arguments. An important new feature of these results is the absense of the near-CMC assumption when the rescaled background metric is in the positive Yamabe class, if the freely specifiable part of the data given by the matter fields (if present) and the traceless-transverse part of the rescaled extrinsic curvature are taken to be sufficiently small. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, giving what are apparently the first non-CMC existence results without the near-CMC assumption. Standard bootstrapping arguments to increase the regularity of the conformal factor are blocked by the use of a weak background metric. In the CMC case, we recover Maxwell's rough solution results as a special case. Our results extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.