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Future global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions $n\geq 4$

Chao Liu, Todd A. Oliynyk, Jinhua Wang

TL;DR

The paper proves global existence and future stability of de Sitter-like solutions to the Einstein--Yang--Mills equations in spacetime dimensions $n\geq 4$ under small perturbations. It introduces a conformal compactification and reformulates the gauge-reduced EYM system as a symmetric hyperbolic Fuchsian system on a conformal de Sitter background, then applies a Beyer-type global existence theorem to obtain uniform bounds and decay toward de Sitter. The analysis combines two gauges (temporal for YM and wave for gravity) with a gauge-transformation framework to bridge local results and global stability, and it provides dimension-dependent parameter choices that certify the required coefficient structure. The results extend Friedrich’s $n=4$ stability to higher dimensions and deliver explicit Sobolev bounds, establishing the robustness of de Sitter-like dynamics under nonlinear EYM perturbations. The work advances stability theory for coupled gauge-field systems in general relativity and supplies a rigorous pathway for global evolution in higher-dimensional cosmologies.

Abstract

We establish the global existence and stability to the future of non-linear perturbation of de Sitter-like solutions to the Einstein-Yang-Mills system in $n \geq 4$ spacetime dimension. This generalizes Friedrich's Einstein-Yang-Mills stability results in dimension $n=4$ [ J Differ Geom 34 (1991), 275-345] to all higher dimensions.

Future global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions $n\geq 4$

TL;DR

The paper proves global existence and future stability of de Sitter-like solutions to the Einstein--Yang--Mills equations in spacetime dimensions under small perturbations. It introduces a conformal compactification and reformulates the gauge-reduced EYM system as a symmetric hyperbolic Fuchsian system on a conformal de Sitter background, then applies a Beyer-type global existence theorem to obtain uniform bounds and decay toward de Sitter. The analysis combines two gauges (temporal for YM and wave for gravity) with a gauge-transformation framework to bridge local results and global stability, and it provides dimension-dependent parameter choices that certify the required coefficient structure. The results extend Friedrich’s stability to higher dimensions and deliver explicit Sobolev bounds, establishing the robustness of de Sitter-like dynamics under nonlinear EYM perturbations. The work advances stability theory for coupled gauge-field systems in general relativity and supplies a rigorous pathway for global evolution in higher-dimensional cosmologies.

Abstract

We establish the global existence and stability to the future of non-linear perturbation of de Sitter-like solutions to the Einstein-Yang-Mills system in spacetime dimension. This generalizes Friedrich's Einstein-Yang-Mills stability results in dimension [ J Differ Geom 34 (1991), 275-345] to all higher dimensions.
Paper Structure (41 sections, 32 theorems, 367 equations)

This paper contains 41 sections, 32 theorems, 367 equations.

Table of Contents

  1. Introduction
  2. Einstein--Yang--Mills equations
  3. de Sitter spacetime
  4. Gauge conditions
  5. Initial data and the constraint equations
  6. Main theorem
  7. Prior and related works
  8. Overview and proof strategy
  9. Main idea
  10. Additional remarks
  11. Fuchsian fields
  12. Derivation of the Fuchsian equation
  13. Outlines
  14. Preliminaries
  15. Abstract indices and tensor conventions
  16. ...and 26 more sections

Key Result

Theorem 1.1

Suppose $\Lambda>0$, $s\in\mathbb{Z}_{>\frac{n+1}{2}}$, and the initial data $\acute{g}_{ab}\in H^{s+1}(\Sigma)$, $\grave{g}_{ab}\in H^s(\Sigma)$, $\acute{A}_a \in H^{s}(\Sigma)$ with $\underline{h}^c{}_a\underline{h}^d{}_b(d\acute{A})_{cd} \in H^{s}(\Sigma)$, and $\grave{A}_a \in H^{s}(\Sigma)$ sat then there exists a unique solution $(\tilde{g}^{ab},\tilde{A}_a)$ to the Einstein--Yang--Mills equ

Theorems & Definitions (63)

  • Remark 1.1
  • Theorem 1.1
  • Lemma 3.1
  • Corollary 3.1
  • proof : Proof of Lemma \ref{['t:rdein']}
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.2
  • ...and 53 more