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Mass-type invariants in the presence of a cosmological constant

Virginia Agostiniani, Stefano Borghini, Lorenzo Mazzieri

Abstract

In this paper, we introduce new mass-type invariants for time-symmetric initial data in space-times obeying the Dominant Energy Condition. When the cosmological constant is positive, these invariants, unlike the total Hawking mass, turn out to be genuinely effective in providing new characterizations of de Sitter solution. From a theoretical standpoint, this opens a new perspective on how one might refine the rigidity statement originally proposed by Min-Oo in his well known conjecture, later refuted by the counterexamples of Brendle, Marques, and Neves.

Mass-type invariants in the presence of a cosmological constant

Abstract

In this paper, we introduce new mass-type invariants for time-symmetric initial data in space-times obeying the Dominant Energy Condition. When the cosmological constant is positive, these invariants, unlike the total Hawking mass, turn out to be genuinely effective in providing new characterizations of de Sitter solution. From a theoretical standpoint, this opens a new perspective on how one might refine the rigidity statement originally proposed by Min-Oo in his well known conjecture, later refuted by the counterexamples of Brendle, Marques, and Neves.
Paper Structure (29 sections, 33 theorems, 380 equations)

This paper contains 29 sections, 33 theorems, 380 equations.

Table of Contents

  1. Introduction and statement of the main results.
  2. The state of the art.
  3. Hawking Mass and Inverse Mean Curvature Flow: failure of a natural strategy.
  4. A potential-theoretic approach: statement of the main results.
  5. Formal limits as $p \to 1^+$: conjectures and further perspectives.
  6. The $1$-harmonic Mass
  7. Organization of the paper.
  8. Monotonicity formulas for $p$-harmonic functions on Riemannian bands
  9. Riemannian bands vs punctured domains.
  10. A selection principle for the structural coefficients
  11. The guideline case $\Lambda=0$.
  12. Green's functions for the $p$-Laplacian on 3-dimensional spaceforms and the selection of $\alpha$
  13. A selection principle for the structural coefficients $\mu$ and $\lambda$: ancient solutions and the small sphere limit.
  14. Global existence and asymptotic behavior of the structural coefficients.
  15. Further properties of the structural coefficients in the case $\Lambda>0$
  16. ...and 14 more sections

Key Result

Theorem 1.1

$(M,g)$ be a complete, asymptotically flat, three-dimensional Riemannian manifold, with nonnegative scalar curvature ${\mathrm R}_g \geq 0$. Then the total ADM-mass of the manifold is nonnegative Moreover, it vanishes if and only if $(M,g)$ is isometric to the flat Euclidean three-space $(\mathbb{R}^3, g_{\mathbb{R}^3})$.

Theorems & Definitions (64)

  • Theorem 1.1: Positive Mass Theorem
  • Theorem 1.2: Scalar Curvature Rigidity for AH Manifolds
  • Conjecture : Min--Oo conjecture
  • Theorem 1.3: Positive Mass Theorem for the Polarized $p$-harmonic Mass
  • Theorem 1.4: Positive Mass Theorem for the $p$-harmonic Total Mass
  • Theorem 1.5: Positive Mass Theorem for Isoperimetric Initial Data
  • Theorem 1.6
  • Theorem 2.1: General Monotonicity Formula
  • Remark 2.2: Geroch-type Formula
  • Remark 2.3: Preserving Connectedness
  • ...and 54 more