arXiv research

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

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

We provide a detailed proof of Hawking's singularity theorem in the regularity class $C^{1,1}$, i.e., for spacetime metrics possessing locally Lipschitz continuous first derivatives. The proof uses recent results in $C^{1,1}$-causality theory and is based on regularisation techniques adapted to the causal structure.

Proves Hawking's theorem for less smooth spacetime metrics.

The paper extends Hawking's singularity theorem to metrics with Hölder continuity and bounded curvature.

Study of spacetimes in cosmology without symmetry assumptions.

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…

This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.

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…

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof o…

We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point set topology.

The Gannon-Lee singularity theorems give well-known restrictions on the spatial topology of singularity-free (i.e., nonspacelike geodesically complete), globally hyperbolic spacetimes. In this paper, we revisit these classic results in the light of recent developments, especially the failure in higher dimensions of a c…

The paper extends Hawking--Page solutions to various spacetimes with singularities.

A singularity theorem based on asymptotic volume growth

The study proves a transverse diameter theorem for Lorentzian foliations.

New singularity concept in GR: volume singularities.

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

A theorem transforms Lorentzian to signature-changing metrics.

The study examines Hawkes processes and their long-term behavior.

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with $C^{1,1}$-metrics with a lower bound on the tim…

We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in $3+1$ dimensions as a hypersurface in $R^{4, 1}$. For the Schwarzschild metric the…

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the ge…

The paper constructs singularities for Lagrangian flow in Gibbons-Hawking spaces with vanishing mean curvature.

We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary condi…

The paper develops Hawkes-based models for LOB and applies them to European, spread, and basket option pricing.

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the …

The Bakry-Emery generalized Ricci tensor arises in scalar-tensor gravitation theories in the conformal gauge known as the Jordan frame. Recent results from the mathematics literature show that standard singularity and splitting theorems that hold when an energy condition is applied in general relativity also hold when …

The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.

We discuss the Ricci-flat `model metrics' on $\mathbb{C}^2$ with cone singularities along the conic $\{zw=1\}$ constructed by Donaldson using the Gibbons-Hawking ansatz over wedges in $\mathbb{R}^3$. In particular we describe their asymptotic behavior at infinity and compute their energies.

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…

A key result in four dimensional black hole physics, since the early 1970s, is Hawking's topology theorem asserting that the cross-sections of an "apparent horizon", separating the black hole region from the rest of the spacetime, are topologically two-spheres. Later, during the 1990s, by applying a variant of Hawking'…

In the present work some generalizations of the Hawking singularity theorems in the context of $f(R)$ theories are presented. The assumptions are of these generalized theorems is that the matter fields satisfy the conditions $\bigg(T_{ij}-\frac{g_{ij}}{2} T\bigg)k^i k^j\geq 0$ for any generic unit time like field, that…

Synthetic framework for null hypersurfaces in non-smooth spacetimes.

In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regards to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for seve…

Study differentially private methods for learning Hawkes processes.

The paper establishes a central limit theorem for estimating the influence parameter in a partially observed Hawkes process system.

The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.

Study of spacelike singularities in spherical spacetimes with scalar matter.

3D metrics get scalar curvature bounds via IMCF.

Paper develops efficient estimator for Hawkes processes using representer theorem.

In this paper, we introduce a new model for the risk process based on general compound Hawkes process (GCHP) for the arrival of claims. We call it risk model based on general compound Hawkes process (RMGCHP). The Law of Large Numbers (LLN) and the Functional Central Limit Theorem (FCLT) are proved. We also study the ma…

Study complex structures of hyperkähler manifolds with infinite type.

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…

We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian-Yau techniques. The proof relies on a construction of good compactifications of Q-Gorenstein deformations of quotient surface singularities…

In this paper we introduce two new Hawkes processes, namely, compound and regime-switching compound Hawkes processes, to model the price processes in limit order books. We prove Law of Large Numbers and Functional Central Limit Theorems (FCLT) for both processes. The two FCLTs are applied to limit order books where we …

In this paper, we study various new Hawkes processes, namely, so-called general compound and regime-switching general compound Hawkes processes to model the price processes in the limit order books. We prove Law of Large Numbers (LLN) and Functional Central Limit Theorems (FCLT) for these processes. The latter two FCLT…

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properl…