arXiv research

Study shows IMP gluing spacetimes are incomplete.

New theorem shows black holes are real, with clearer event horizon.

A singularity theorem based on asymptotic volume growth

Researchers prove charged Penrose inequality and positive mass theorem for specific manifold types.

Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.

The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.

Cosmological singularity theorems such as that of Hawking and Penrose assume local curvature conditions as well as global ones like the existence of a compact (achronal) slice. Here, we prove a new singularity theorem for chronological spacetimes that satisfy what we call a `past null focusing' condition. Such a condit…

New singularity concept in GR: volume singularities.

We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity $C^{1,1}$. The proof is based on regularisation techniques, combined with recent results in low regularity causality theory.

The Penrose-Kauffman polynomial connects knot theory to graph coloring.

The paper proves a Hawking-type singularity theorem using worldvolume quantum strong energy inequalities.

We give, via elementary methods, explicit formulas for the ADM mass which allow us to conclude the positive mass theorem and Penrose inequality for a class of graphical manifolds which includes, for instance, that ones with flat normal bundle.

Study Penrose inequality for metrics with singular sets.

We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.

In a recent paper, Eichmair, Galloway and Pollack have proved a Gannon-Lee-type singularity theorem based on the existence of marginally outer trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic spacetimes. However, one might wonder whether the corresponding incomplete geodesics could still …

In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.

In this paper we prove a rigidity result for the equality case of the Penrose inequality on $3$-dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners. Our result also has deep connections with the equality cases of Theorem 1 in \cite{Miao2} and Theorem 1.1 in \cite{LM}.

New proof of Riemannian Penrose Inequality for manifolds with corners

The Spencer cohomology of certain graded Lie superalgebras are completely computed. This cohomology is interpreted as analogs of Riemann and Penrose tensors on supermanifolds. The results make it manifest that there is no simple generalization of Borel-Weil-Bott's theorem for Lie superalgebras.

In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes fro…

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…

New metrics derived from geodesics simplify semi-Riemannian geometry.

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, an…

We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to R^3. We establish a generalization of the Penrose singularity theorem…

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities" in Riemannian manifolds, general…

New proof of Positive Mass Theorem using Green's function and monotonicity formula.

Researchers prove a Penrose inequality for spacetime with specific conditions.

The paper proves a Penrose inequality involving quasi-local mass and outer trapped surfaces.

The paper proves density and positive mass theorems for incomplete manifolds.

Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.

Synthetic proof of Gannon-Lee theorem for spacetimes.

Researchers prove a 30-year-old cosmological conjecture about spacetime.

We use the inverse mean curvature flow to establish Penrose-type inequalities for time-symmetric Einstein-Maxwell initial data sets which can be suitably embedded as a hypersurface in Euclidean space $\mathbb R^{n+1}$, $n\geq 3$. In particular, we prove a positive mass theorem for this class of charged black holes. As …

The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost min…

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main the…

We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature $Ric_f^m$ in timelike directions. We prove that they still hold when $m$ is finite, and when $m$ is infinite, they hold under the additional assumptio…

New mass and staticity concepts derived from weighted curvature maps.

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss-Bonnet-Chern mass $m_2$ for the graphic manifold with flat normal bundle.

This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries a…

The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.

We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we …

Study shows formation of Kerr black holes with complete apparent horizons and proves Penrose inequalities.

The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.

We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus provin…

The noncompact Yamabe flow can lead to incomplete metrics over infinite time.

In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.

Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface $Σ$, we observe that …

In a recent paper, Eichmair, Galloway and Pollack have proved a Gannon-Lee-type singularity theorem based on the existence of marginally outer trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic spacetimes. This result requires that the MOTS be generic in a suitable sense. In the same spirit…