Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends

TL;DR

This paper constructs new initial data sets for Einstein's equations with positive cosmological constant, featuring exactly Delaunay ends, expanding the variety of known solutions with specific asymptotic geometries.

Contribution

It introduces a method to generate initial data with precise Delaunay ends, including non-spherical cross-sections, for the vacuum Einstein equations with positive cosmological constant.

Findings

Constructed large families of initial data with Delaunay ends.

Produced complete, constant positive scalar curvature metrics with non-globally Delaunay ends.

Enabled new compact initial data sets via gluing techniques.

Abstract

We construct large families of initial data sets for the vacuum Einstein equations with positive cosmological constant which contain exactly Delaunay ends; these are non-trivial initial data sets which coincide with those for the Kottler-Schwarzschild-de Sitter metrics in regions of infinite extent. From the purely Riemannian geometric point of view, this produces complete, constant positive scalar curvature metrics with exact Delaunay ends which are not globally Delaunay. The ends can be used to construct new compact initial data sets via gluing constructions. The construction provided applies to more general situations where the asymptotic geometry may have non-spherical cross-sections consisting of Einstein metrics with positive scalar curvature.