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Foliations by critical surfaces of the Hawking energy in asymptotically flat initial data sets

Alejandro Peñuela Diaz

TL;DR

This work constructs global foliations at infinity by area-constrained Hawking surfaces in asymptotically Schwarzschild initial data and establishes key physical properties of the Hawking energy along these leaves, including nonnegativity, a large-sphere limit to the ADM energy, and, under an integral condition, monotonicity. It extends the Willmore/Willmore-like foliation framework to dynamical data using a Lyapunov-Schmidt reduction, and analyzes the center of mass associated with the foliation, providing a center formula that may differ from the STCMC center in general. The results are robust under harmonic and York asymptotics and include a rigidity statement characterizing Minkowski space when the Hawking energy vanishes on a leaf. Overall, Hawking surfaces emerge as a robust quasi-local energy tool with well-controlled asymptotics and center-of-mass behavior in dynamical spacetimes.

Abstract

Area-constrained critical surfaces for the Hawking quasi-local energy ("Hawking surfaces") provide a natural setting for that energy: they enjoy positivity and rigidity properties. We construct large-scale foliations at infinity by Hawking surfaces in asymptotically Schwarzschild initial data sets. Using a Lyapunov-Schmidt reduction within a Willmore-foliation framework, we prove existence and uniqueness of the foliation and study its coordinate center. Under the dominant energy condition, we show that along the leaves of the foliation, the Hawking energy is positive and converges to the ADM energy in the large-sphere limit; moreover, subject to an explicit integral constraint, it is monotone along the foliation. Under weaker assumptions we construct an on-center family of Hawking surfaces that, while not necessarily a foliation, still enjoys positivity and the large-sphere limit. Finally, we obtain a rigidity statement and verify that our hypotheses hold in a broad class of data, initial data sets with harmonic or York asymptotics, thereby demonstrating the robustness of Hawking surfaces as a quasi-local energy tool in dynamical spacetimes.

Foliations by critical surfaces of the Hawking energy in asymptotically flat initial data sets

TL;DR

This work constructs global foliations at infinity by area-constrained Hawking surfaces in asymptotically Schwarzschild initial data and establishes key physical properties of the Hawking energy along these leaves, including nonnegativity, a large-sphere limit to the ADM energy, and, under an integral condition, monotonicity. It extends the Willmore/Willmore-like foliation framework to dynamical data using a Lyapunov-Schmidt reduction, and analyzes the center of mass associated with the foliation, providing a center formula that may differ from the STCMC center in general. The results are robust under harmonic and York asymptotics and include a rigidity statement characterizing Minkowski space when the Hawking energy vanishes on a leaf. Overall, Hawking surfaces emerge as a robust quasi-local energy tool with well-controlled asymptotics and center-of-mass behavior in dynamical spacetimes.

Abstract

Area-constrained critical surfaces for the Hawking quasi-local energy ("Hawking surfaces") provide a natural setting for that energy: they enjoy positivity and rigidity properties. We construct large-scale foliations at infinity by Hawking surfaces in asymptotically Schwarzschild initial data sets. Using a Lyapunov-Schmidt reduction within a Willmore-foliation framework, we prove existence and uniqueness of the foliation and study its coordinate center. Under the dominant energy condition, we show that along the leaves of the foliation, the Hawking energy is positive and converges to the ADM energy in the large-sphere limit; moreover, subject to an explicit integral constraint, it is monotone along the foliation. Under weaker assumptions we construct an on-center family of Hawking surfaces that, while not necessarily a foliation, still enjoys positivity and the large-sphere limit. Finally, we obtain a rigidity statement and verify that our hypotheses hold in a broad class of data, initial data sets with harmonic or York asymptotics, thereby demonstrating the robustness of Hawking surfaces as a quasi-local energy tool in dynamical spacetimes.

Paper Structure

This paper contains 11 sections, 34 theorems, 242 equations.

Table of Contents

  1. Introduction and Main Results
  2. Main results
  3. Foliation construction and large-sphere limit
  4. Foliation construction
  5. Large-sphere limit
  6. Center of the foliation
  7. Nonnegativity, monotonicity and rigidity
  8. Nonnegativity along the foliation
  9. Monotonicity
  10. Rigidity on the foliation
  11. Density of Harmonic asymptotics

Key Result

Theorem 2.1

Let $(M, g)$ be $\mathcal{C}^4$-asymptotic to Schwarzschild with mass $m > 0$ and suppose that the scalar curvature $\mathrm{Sc}$ satisfies There exists a compact set $K \subset M$, a number $\lambda_0 > 0$, and stable, on-center, area-constrained critical Willmore spheres $\Sigma(\lambda)$, $\lambda \in (0, \lambda_0)$, with parameter $\lambda$ such that $M \setminus K$ is foliated by the family

Theorems & Definitions (68)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4: Coordinate center of a foliation
  • Theorem 2.1: eichko
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • ...and 58 more