Positive Mass Theorem Research Articles

Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

AbstractLet (M, g) be an asymptotically flat Riemannian 3‐manifold with nonnegative scalar curvature and positive mass. We show that each leaf of the canonical foliation of the end of (M, g) through stable constant mean curvature spheres encloses more volume than any other surface of the same area.Unlike all previous characterizations of large solutions of the isoperimetric problem, we need no asymptotic symmetry assumptions beyond the optimal conditions for the positive mass theorem. This generality includes examples where global uniqueness of the leaves of the canonical foliation as stable constant mean curvature spheres fails dramatically.Our results here resolve a question of G. Huisken on the isoperimetric content of the positive mass theorem. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

A positive mass theorem for static causal fermion systems

Asymptotically flat static causal fermion systems are introduced. Their total mass is defined as a limit of surface layer integrals which compare the measures describing the asymptotically flat spacetime and a vacuum spacetime near spatial infinity. Our definition does not involve any regularity assumptions; it even applies to singular or generalized "quantum" spacetimes. A positive mass theorem is proven. Our methods and results explain why and how the causal action principle incorporates the nonlinear effects of gravity for static systems.

Limiting Case of the Spin Hypersurface Dirac Operator arising in the positive mass theorem for black holes

AbstractIn this paper, we give an explicit form of the scalar curvaure for the limiting case of the eigenvalue of the hypersurface Dirac operator which arises in the positive mass theorem for black holes. Then, we show that the hypersurface is an Einstein.

On the Negativity of Ricci Curvatures of Complete Conformal Metrics

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases.

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.

A quasi-local Penrose inequality for the quasi-local energy with static references

The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the cosmic censorship conjecture in general relativity. In [14], Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy with reference in the Schwarzschild manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy with reference in any spherically symmetric static spacetime.

On the Minimizers of Curvature Functionals in Asymptotically Flat Manifolds

In this paper, we study the minimizers of curvature functionals (Willmore functional and extrinsic energy functional) subject to an area constraint in asymptotically flat manifolds. Under some certain conditions, we prove that such minimizers exist. Besides the surface theory related to the Willmore functional, the proofs also rely on the inverse mean curvature flow developed by Huisken and Ilmanen, on the positive mass theorem due to Schoen–Yau, and on the positive mass theorem for asymptotically flat manifolds with corners due to Miao and Shi–Tam. Our results may be of some interest in the General Relativity.

Spacetime Positive Mass Theorems for Initial Data Sets with Non-Compact Boundary

AbstractIn this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is light-like. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

A positive mass theorem for manifolds with boundary

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.

Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper, we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in \(L^p\), \(1\le p<2\). In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.

The Penrose Inequality and Positive Mass Theorem with Charge for Manifolds with Asymptotically Cylindrical Ends

We establish the charged Penrose inequality for time symmetric initial data sets having an outermost minimal surface boundary and finitely many asymptotically cylindrical ends, with an appropriate rigidity statement. This is accomplished by a doubling argument based on the work of Weinstein and Yamada, and a subsequent application of the ordinary charged Penrose inequality as established by Khuri, Weinstein, and Yamada. Furthermore, the techniques used in the aforementioned proof allow for the proof of the positive mass theorem with charge for such manifolds.

The Positive Mass Theorem for Non-spin Manifolds with Distributional Curvature

We prove the positive mass theorem for manifolds with distributional curvature which have been studied in \cite{Lee2015} without spin condition. In our case, the manifold $M$ has asymptotically flat metric $g\in C^0\bigcap W^{1,p}_{-q}$, $p>n$, $q>\frac{n-2}{2}$. We show that the generalized ADM mass $m_{ADM}(M,g)$ is non-negative as long as $q=n-2$, and $g$ has non-negative distributional scalar curvature, bounded curvature in the Alexandrov sense with its distributional Ricci curvature belonging to certain weighted Lebesgue space and some extra conditions.

NNSC-Cobordism of Bartnik Data in High Dimensions

In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are $(n-1)$-dimensional Bartnik data $\big(\Sigma_i ^{n-1}, \gamma_i, H_i\big)$, $i=1,2$, NNSC-cobordant? (i.e., there is an $n$-dimensional compact Riemannian manifold $\big(\Omega^n, g\big)$ with scalar curvature $R(g)\geq 0$ and the boundary $\partial \Omega=\Sigma_{1} \cup \Sigma_{2}$ such that $\gamma_i$ is the metric on $\Sigma_i ^{n-1}$ induced by $g$, and $H_i$ is the mean curvature of $\Sigma_i$ in $\big(\Omega^n, g\big)$). If $\big(\mathbb{S}^{n-1},\gamma_{\rm std},0\big)$ is positive scalar curvature (PSC) cobordant to $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$, where $\big(\mathbb{S}^{n-1}, \gamma_{\rm std}\big)$ denotes the standard round unit sphere then $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$ admits an NNSC fill-in. Just as Gromov's conjecture is connected with positive mass theorem, our problems are connected with Penrose inequality, at least in the case of $n=3$. Our third problem is on $\Lambda\big(\Sigma^{n-1}, \gamma\big)$ defined below.

Stability of the positive mass theorem for axisymmetric manifolds

Away from the central axis, we prove the stability of the Positive Mass Theorem in the $W^{1,p}$ sense for asymptotically flat axisymmetric manifolds with nonnegative scalar curvature satisfying some additional technical assumptions. We also derive estimates for the volumes of regions, the areas of axisymmetric surfaces, and the distances between points within the manifolds.

Bartnik mass via vacuum extensions

We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].

Positive energy theorem for asymptotically anti-de Sitter spacetimes with distributional curvature

We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

Mass Rigidity for Hyperbolic Manifolds

We prove the rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality $$p_0=\sqrt{p_1^2+\cdots + p_n^2}$$ holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds (Andersson and Dahl in Ann Glob Anal Geom 16(1):1–2, 1998; Chruściel and Herzlich in Pac J Math 212(2):231–264, 2003; Min-Oo in Math Ann 285(4):527–539; 1989, Wang in J Differ Geom 57(2):273–299, 2001) or under special asymptotics (Andersson et al. in Ann. Henri Poincaré 9(1):1–33, 2008).

Equality in the Spacetime Positive Mass Theorem

We affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight. Namely, if an asymptotically flat initial data set satisfies the dominant energy condition and has $$E=|P|$$, then $$E=|P|=0$$, where (E, P) is the ADM energy-momentum vector. The dimensional restriction can be removed if we assume the positive mass inequality holds. Previously the result was only known for spin manifolds (Beig and Chruściel in J Math Phys37(4):1939–1961, 1996; Chruściel and Maerten in J Math Phys 47(2), 2006).

Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In $3$-dimension, Shi-Tam’s result is known to be equivalent to the Riemannian positive mass theorem. In this paper, we provide a supplement to Shi–Tam’s result by including the boundary effect of minimal hypersurfaces. More precisely, given a compact manifold $\Omega$ with nonnegative scalar curvature, assuming its boundary consists of two parts, $\Sigma_H$ and $\Sigma_O$, where $\Sigma_H$ is the union of all closed minimal hypersurfaces in $\Omega$ and $\Sigma_O$ is assumed to be isometric to a suitable 2-convex hypersurface $\Sigma$ in a spatial Schwarzschild manifold of mass $m$, we establish an inequality relating $m$, the area of $\Sigma_H$, and two weighted total mean curvatures of $\Sigma_O$ and $\Sigma$. In $3$-dimension, our inequality has implications to isometric embedding and quasi-local mass problems. In a relativistic context, the result can be interpreted as a quasi-local mass type quantity of $\Sigma_O$ being greater than or equal to the Hawking mass of $\Sigma_H$. We further analyze the limit of this quantity associated with suitably chosen isometric embeddings of large spheres in an asymptotically flat $3$-manifold $M$ into a spatial Schwarzschild manifold. We show that the limit equals the ADM mass of $M$. It follows that our result on the compact manifold $\Omega$ is equivalent to the Riemannian Penrose inequality.

Lower semicontinuity of the ADM mass in dimensions two through seven

The semicontinuity phenomenon of the ADM mass under pointed (i.e., local) convergence of asymptotically flat metrics is of interest because of its connections to nonnegative scalar curvature, the positive mass theorem, and Bartnik's mass-minimization problem in general relativity. In this paper, we extend a previously known semicontinuity result in dimension three for $C^2$ pointed convergence to higher dimensions, up through seven, using recent work of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality of H. Bray and D. Lee). For a technical reason, we restrict to the case in which the limit space is asymptotically Schwarzschild. In a separate result, we show that semicontinuity holds under weighted, rather than pointed, $C^2$ convergence, in all dimensions $n \geq 3$, with a simpler proof independent of the positive mass theorem. Finally, we also address the two-dimensional case for pointed convergence, in which the asymptotic cone angle assumes the role of the ADM mass.