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Sobolev regularity of compactified 3-manifolds and the ADM Center of Mass

Rodrigo Avalos

Abstract

In this paper, we address the existence of preferred asymptotic coordinates on asymptotically Euclidean (AE) manifolds $(M^3,g)$ such that $g$ admits an asymptotically Schwarzschildian first order expansion, based purely on a priori geometric conditions, which will then be used to establish geometric criteria guaranteeing the convergence of the ADM center of mass (COM). This question is analysed relating it to the study of the regularity of conformal compactifications of such manifolds, which is itself explored via elliptic theory for operators with coefficients of very limited regularity. With these related problems in mind, we shall first establish regularity properties of $L^{q'}$-solutions associated to the operator $Δ_{\hat{g}}:W^{k,p}(S_2\hat{M})\to W^{k-2,p}(S_2\hat{M})$, for $k=1,2$ and $1<p\leq q$, where $\hat{M}^n$ is a closed manifold, $\hat{g}\in W^{2,q}(\hat{M})$, with $q>\frac{n}{2}$ and $S_2\hat{M}$ denotes the bundle of symmetric $(0,2)$-tensor fields. Appealing to these results, we establish Sobolev regularity of conformally compactified AE 3-manifolds via the decay of the Cotton tensor, improving on previous results. This allows us to construct preferred asymptotic coordinates on such AE manifolds where the metric has a first order Schwarzschildian expansion, which in turn will allow us to address a version of a conjecture posed by C. Cederbaum and A. Sakovich concerning the convergence of the COM of such manifolds.

Sobolev regularity of compactified 3-manifolds and the ADM Center of Mass

Abstract

In this paper, we address the existence of preferred asymptotic coordinates on asymptotically Euclidean (AE) manifolds such that admits an asymptotically Schwarzschildian first order expansion, based purely on a priori geometric conditions, which will then be used to establish geometric criteria guaranteeing the convergence of the ADM center of mass (COM). This question is analysed relating it to the study of the regularity of conformal compactifications of such manifolds, which is itself explored via elliptic theory for operators with coefficients of very limited regularity. With these related problems in mind, we shall first establish regularity properties of -solutions associated to the operator , for and , where is a closed manifold, , with and denotes the bundle of symmetric -tensor fields. Appealing to these results, we establish Sobolev regularity of conformally compactified AE 3-manifolds via the decay of the Cotton tensor, improving on previous results. This allows us to construct preferred asymptotic coordinates on such AE manifolds where the metric has a first order Schwarzschildian expansion, which in turn will allow us to address a version of a conjecture posed by C. Cederbaum and A. Sakovich concerning the convergence of the COM of such manifolds.
Paper Structure (19 sections, 64 theorems, 564 equations)

This paper contains 19 sections, 64 theorems, 564 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Geometric conventions
  5. Analytic tools
  6. Sobolev spaces
  7. Mapping properties of $\mathcal{L}^2(W^{2,q})$-operators - $q>\frac{n}{2}$
  8. AE Manifolds
  9. ADM definitions
  10. Semi-Fredholm properties of $\mathcal{L}^2(W^{2,q})$-Elliptic operators
  11. Regularity theory for $\Delta_g$ on $S_2M$
  12. $W^{2,p}$-regularity theory
  13. $W^{1,p}$-regularity theory
  14. Conformal properties of low regularity closed Riemannian manifolds
  15. Compactification of AE 3-manifolds - improved regularity
  16. ...and 4 more sections

Key Result

Theorem A

Let $(M^n,g)$ be a closed Riemannian manifold with $g\in W^{2,q}(M)$, $q>\frac{n}{2}$, and $n\geq 3$. Then, given $1< p\leq q$, the Laplace-Beltrami operator is Fredholm of index zero, $\mathrm{Ker}(\Delta_g\vert_{W^{2,p}})\subset W^{2,q}(M)$ and the following regularity implication follows:

Theorems & Definitions (142)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Conjecture 1.1: Cederbaum-Sakovich CederbaumSakovich
  • Conjecture 1.2: Weak version of Conjecture \ref{['CSConjecture']}
  • Theorem E
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • ...and 132 more