Vacuum initial data with minimal decay and borderline decay

TL;DR

The work tackles vacuum initial data for the Einstein equations at minimal and borderline decay by extending Mao-Tao’s conic solution-operator framework to a spacelike setting. It constructs cone-supported seed data and, via an outgoing right inverse $S$ on $b$-Sobolev spaces, solves the nonlinear constraint system to obtain nontrivial $(s,q)$--asymptotically flat initial data for all $s\in[1,3)$ and $q\in\mathbb{N}$. The main result exhibits sharp decay thresholds: the seed data lie in $H_b^{q+1,(s-4)/2}\times H_b^{q,(s-2)/2}$ but not in any stronger weight class for $s'>s$, and in the borderline case $s=1$ the data remain exterior to global stability results while fitting Shen24’s exterior stability framework. This advances the understanding of how minimal and borderline decay interact with the Einstein constraint equations and the conic localization program, producing initial data compatible with exterior stability (and highlighting limits for global stability) in the Minkowski regime.

Abstract

In this note, we show that the conical solution-operator method of Mao-Tao in [Localized initial data for Einstein equations] applies to a simple construction of vacuum asymptotically flat initial data at minimal and borderline decay thresholds, corresponding to the global and exterior stability of Minkowski spacetime proved by the first named author in [Global stability of Minkowski spacetime with minimal decay] and [Exterior stability of Minkowski spacetime with borderline decay].

Theorems & Definitions (31)

• Definition 1.1
• Theorem 1.2: Global stability of Minkowski with minimal decay Shen23
• Theorem 1.3: Exterior stability of Minkowski with borderline decay Shen24
• Definition 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 2.1: Conical fundamental kernels MaoTao
• Remark 2.2
• Definition 3.1
• Remark 3.2