Existence and Blowup Results for Asymptotically Euclidean Initial Data Sets Generated by the Conformal Method

TL;DR

This paper establishes criteria for the existence of solutions to Einstein constraint equations using the conformal method, analyzes blowup behavior near non-solution seed data, and constructs AE seed data with specific geometric properties.

Contribution

It introduces an admissibility criterion based on scalar curvature, studies blowup phenomena, and constructs AE seed data with Yamabe nonpositive metrics and zero mean curvature.

Findings

Admissibility criterion for seed data based on scalar curvature

Blowup analysis as seed data approach non-solution sets

Existence of AE seed data with Yamabe nonpositive metrics

Abstract

For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data sets approach sets for which no solutions exist. We also prove that there are AE seed data sets which include a Yamabe nonpositive metric and lead to solutions of the conformal constraints. These data sets allow the mean curvature function to have zeroes.