Non-constant mean curvature trumpet solutions for the Einstein constraint equations

TL;DR

This paper establishes the existence of a broad class of initial data solutions for the vacuum Einstein equations with multiple ends, under mild mean curvature conditions, expanding the understanding of initial data configurations in general relativity.

Contribution

It introduces a new class of initial data with multiple ends and mild mean curvature conditions, broadening the scope of known solutions in Einstein's equations.

Findings

Existence of initial data with multiple ends proven.

Initial data sets are asymptotically Euclidean or conformally cylindrical/periodic.

Mild conditions on mean curvature are sufficient for existence.

Abstract

We prove the existence of a large class of initial data for the vacuum Einstein equations which possess a finite number of asymptotically Euclidean and asymptotically conformally cylindrical or periodic ends. Aside from being asymptotically constant, only mild conditions on the mean curvature of these initial data sets are imposed.