Convex bodies: the brunn–minkowski theory

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

The present status of the quasi-local mass-energy-momentum and angular momentum constructions in general relativity is reviewed. First the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities are recalled. Then the various specific constructions and their properties (both successes and defects) are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned. This review is based on the talks given at the Erwin Schrodinger Institute, Vienna, in July 1997, at the Universitat Tubingen, in May 1998, and at the National Center for Theoretical Sciences in Hsinchu and at the National Central University, Chungli, Taiwan, in July 2000.

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Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article

The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.

/pdf/quasi-local-energy-momentum-and-angular-momentum-in-general-wmf4x6598j.pdf

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity.

In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem.

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Positive Mass Theorem and the Boundary Behaviors of Compact Manifolds with Nonnegative Scalar Curvature

We study a class of non-smooth asymptotically flat manifolds on which metrics fails to be $C^1$ across a hypersurface $\Sigma$. We first give an approximation scheme to mollify the metric, then we prove that the Positive Mass Theorem still holds on these manifolds if a geometric boundary condition is satisfied by metrics separated by $\Sigma$.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2002/0006/0006/ATMP-2002-0006-0006-a004.pdf

Positive mass theorem on manifolds admitting corners along a hypersurface

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

On the volume functional of compact manifolds with boundary with constant scalar curvature

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo’s conjecture such as that of Brendle–Marques–Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

Deformation of scalar curvature and volume

Let R be a constant. Let M R γ be the space of smooth metrics g on a given compact manifold Ω n (n > 3) with smooth boundary Σ such that g has constant scalar curvature R and g|Σ is a fixed metric γ on Σ. Let V(g) be the volume of g ∈ M R γ . In this work, we classify all Einstein or conformally flat metrics which are critical points of V(·) in M R γ .

/pdf/einstein-and-conformally-flat-critical-metrics-of-the-volume-2ets9h1g7i.pdf

Einstein and conformally flat critical metrics of the volume functional

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ3, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

Pengzi Miao

Papers

Positive mass theorem on manifolds admitting corners along a hypersurface

On the volume functional of compact manifolds with boundary with constant scalar curvature

Deformation of scalar curvature and volume

Einstein and conformally flat critical metrics of the volume functional

On the capacity of surfaces in manifolds with nonnegative scalar curvature