• Corpus ID: 232014635

Spacetime Harmonic Functions and Applications to Mass

  title={Spacetime Harmonic Functions and Applications to Mass},
  author={Hubert L. Bray and Sven Hirsch and Demetre Kazaras and Marcus A. Khuri and Yiyue Zhang},
In the pioneering work of Stern [73], level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, are similarly effective in treating geometric inequalities involving the ADM mass. In this paper, we survey recent results in this context, focusing on applications of spacetime harmonic functions to the… 

Figures from this paper

On a spacetime positive mass theorem with corners

In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump

A perturbation of spacetime Laplacian equation

We study a perturbation ∆u+ P |∇u| = h|∇u|, of spacetime Laplacian equation in an initial data set (M,g, p) where P is the trace of the symmetric 2-tensor p and h is a smooth function. Stern [Ste19]

The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets with Toroidal Infinity and Related Rigidity Results

We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying

Spectral Torical Band Inequalities and Generalizations of the Schoen-Yau Black Hole Existence Theorem

Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain

Dihedral rigidity for cubic initial data sets

In this paper we pose and prove a spacetime version of Gromov’s dihedral rigidity theorem ([17],[24],[25]) for cubes when the dimension is 3 by studying the level sets of spacetime harmonic functions

Hawking Mass Monotonicity for Initial Data Sets

We introduce new systems of PDE on initial data sets $(M,g,k)$ whose solutions model double-null foliations. This allows us to generalize Geroch's monotonicity formula for the Hawking mass under



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

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

Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the

The spacetime positive mass theorem in dimensions less than eight

We prove the spacetime positive mass theorem in dimensions less than eight. This theorem states that for any asymptotically flat initial data set satisfying the dominant energy condition, the ADM


We prove the Riemannian Penrose Conjecture, an important case of a con- jecture (41) made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of

Positive scalar curvature and minimal hypersurface singularities

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all

Geometric Inequalities for Quasi-Local Masses

In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach,

Conserved quantities on asymptotically hyperbolic initial data sets

In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total

The Riemannian Penrose Inequality with Charge for Multiple Black Holes

We present the outline of a proof of the Riemannian Penrose in- equality with charge r ≤ m + m 2 − q 2 ,w hereA =4 πr 2 is the area of the outermost apparent horizon with possibly multiple connected

Deformations of Axially Symmetric Initial Data and the Mass-Angular Momentum Inequality

We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically

A Jang Equation Approach to the Penrose Inequality

We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime