Sur les difféomorphismes de la sphère de dimension trois (Г[4]=0)

Total Mean Curvature

We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to $\infty$ which include the scaling of any fixed coordinate-convex surface.

Mass of asymptotically flat 3-manifolds with boundary

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Neumann problem for singular degenerate parabolic equations

. We provide a new proof of the Riemannian Penrose inequality by means of a monotonicity formula holding along the level sets of the p –capacitary potential of the horizon boundary, in any asymptotically ﬂat 3–manifold with nonnegative scalar curvature.

Riemannian Penrose inequality via Nonlinear Potential Theory

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth `spacetime harmonic functions' in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved.

Spacetime Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Initial Data for the Einstein Equations

We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application, we obtain an inequality relating the mass and harmonic functions that generalizes H. Bray's mass-capacity inequality in his proof of the Riemannian Penrose conjecture.

A Positive Mass Theorem for Manifolds with Boundary

. The Theorem of Bonnet–Myers implies that manifolds with topology M n − 1 × S 1 do not admit a metric of positive Ricci curvature, while the resolution of Geroch’s conjecture implies that the torus T n does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so called m -intermediate curvature), and use stable weighted slicings to show that for n ≤ 7 the manifolds N n = M n − m × T m do not admit a metric of positive m -intermediate curvature.

A generalization of Geroch's conjecture

The rigidity of the spacetime positive mass theorem states that an initial data set (M, g, k) satisfying the dominant energy condition with vanishing mass can be isometrically embedded into Minkowski space. This has been established by Beig–Chruściel and Huang–Lee under additional decay assumptions for the energy and momentum densities $$\mu $$ and J. In this note, we give a new and elementary proof in dimension 3 which removes these additional decay assumptions. Our argument uses spacetime harmonic functions and Liouville’s theorem. We also provide an alternative proof based on the Killing development of (M, g, k). 

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Sven Hirsch

Papers

Spacetime Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Initial Data for the Einstein Equations

Mass of asymptotically flat 3-manifolds with boundary

A Positive Mass Theorem for Manifolds with Boundary

A generalization of Geroch's conjecture

The Case of Equality for the Spacetime Positive Mass Theorem