Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics

TL;DR

This paper proves the existence of solutions to Einstein's constraint equations with non-constant mean curvature on closed manifolds, using innovative topological and barrier methods that relax previous derivative restrictions.

Contribution

It introduces new fixed-point and super-solution constructions that allow for arbitrary mean extrinsic curvature without derivative size restrictions.

Findings

Existence of solutions for non-constant mean curvature on closed manifolds.

Methods applicable to weak solutions and manifolds with boundary.

First results free of smallness conditions on derivatives of mean extrinsic curvature.

Abstract

In this article we develop some new existence results for the Einstein constraint equations using the Lichnerowicz-York conformal rescaling method. The mean extrinsic curvature is taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant. The rescaled background metric belongs to the positive Yamabe class, and the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are taken to be sufficiently small, with the matter energy density not identically zero. Using topological fixed-point arguments and global barrier constructions, we then establish existence of solutions to the constraints. Two recent advances in the analysis of the Einstein constraint equations make this result possible: A new type of topological fixed-point argument without smallness conditions on spatial derivatives of the mean extrinsic curvature, and a new construction of global super-solutions for the Hamiltonian constraint that is similarly free of such conditions on the mean extrinsic curvature. For clarity, we present our results only for strong solutions on closed manifolds. However, our results also hold for weak solutions and for other cases such as compact manifolds with boundary; these generalizations will appear elsewhere. The existence results presented here for the Einstein constraints are apparently the first such results that do not require smallness conditions on spatial derivatives of the mean extrinsic curvature.