arXiv research

Analyzing static solutions in Finsler gravity, extending known results.

The study finds static solutions in symplectic curvature flow in 4D.

In a seminal paper of 1917, H. Weyl presented a remarkable reduction of the static axisymmetric vacuum Einstein equations, serving as a relatively straightforward technique to generate and explore new solutions. Weyl's reduction was used by Myers in 1987, and independently by Korotkin-Nicolai in 1994, to construct a ne…

We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean space, there exists a unique AF solution to the static vacuum equations realizin…

New black hole solutions cannot be rotated without breaking their structure.

Static spacetimes are stable attractors in a flow equation.

The paper confirms the existence of 5D regular static vacuum solutions with multiple black holes and Kasner asymptotics.

In this paper, we present two observations about static spherically symmetric solutions of the Einstein-Klein-Gordon equations. The first is a comment extending the well-known result of the existence of static states (i.e. standing wave solutions) of the Einstein-Klein-Gordon equations. The second more important observ…

We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions whi…

We construct infinite-dimensional families of non-singular static space times, solutions of the vacuum Einstein-Maxwell equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with the usual AdS conformal structure at conformal infinity.

We show that the recent work of Lee [23] implies existence of a large class of new singularity-free strictly static Lorentzian vacuum solutions of the Einstein equations with a negative cosmological constant. This holds in all space-time dimensions greater than or equal to four, and leads both to strictly static soluti…

In this paper we study non-singular vacuum static space-times with non-zero cosmological constant. We introduce new integral quantities, and under suitable assumptions we prove their monotonicity along the level set flow of the static potential. We then show how to use these properties to derive a number of sharp geome…

In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum…

The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson et al, and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. Between this article and its sequel we …

We consider the optimal solutions to the trade execution problem in the two different classes of i) fully adapted or adaptive and ii) deterministic or static strategies, comparing them. We do this in two different benchmark models. The first model is a discrete time framework with an information flow process, dealing w…

A complete characterization is obtained of the asymptotic behavior of solutions of the static vacuum Einstein equations which have a (pseudo)-compact horizon or boundary and are complete away from the boundary. It is proved that the time-symmetric space-like hypersurface has only finitely many ends, each of which is ei…

We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary.

This is the second article of a series or two, proving a generalisation of the uniqueness theorem of the Schwarzschild solution. The theorem to be shown classifies all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without any further assumptio…

We investigate Bartnik's static metric extension conjecture under the additional assumption of axisymmetry of both the given Bartnik data and the desired static extensions. To do so, we suggest a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum e…

The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.

In this paper we utilize symmetries in order to exhibit exact solutions to Einstein's equation of a perfect fluid on a static manifold all of whose spatial factor belongs to the conformal class of a Riemannian space of constant curvature.

In this paper we propose and discuss a notion of mass for compact static metrics with positive cosmological constant. As a consequence, we characterise the de Sitter solution as the only static vacuum metric with zero mass. Finally, we show how to adapt our analysis to the case of negative cosmological constant, leadin…

In incomplete financial markets not every contingent claim can be replicated by a self-financing strategy. The risk of the resulting shortfall can be measured by convex risk measures, recently introduced by Föllmer, Schied (2002). The dynamic optimization problem of finding a self-financing strategy that minimizes the …

Ancient solutions and translators identified for Lagrangian flow.

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the l…

The paper explores rigidity and splitting theorems for sub-static spaces with minimal hypersurfaces.

We show that in any spacetime dimension $D\ge 4$, degenerate components of the event horizon do not exist in static vacuum configurations with positive cosmological constant. We also show that without a cosmological constant asymptotically flat solutions cannot possess a degenerate horizon component. Several independen…

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik …

This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part, we have established a positive mass statement, characterising the de Sitter solution as…

Study on stellar models' topology and mass using minimal surfaces.

We compare optimal static and dynamic solutions in trade execution. An optimal trade execution problem is considered where a trader is looking at a short-term price predictive signal while trading. When the trader creates an instantaneous market impact, it is shown that transaction costs of optimal adaptive strategies …

We construct a large class of new singularity-free static Lorentzian four-dimensional solutions of the vacuum Einstein equations with a negative cosmological constant. The new families of metrics contain space-times with, or without, black hole regions. Two uniqueness results are also established.

In this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity. In the case where the static potential is bounded, we introduce new quantities which are proven to be monotone along the level set flow of the potential function. We then show how to use these prope…

We analyze (the harmonic map representation of) static solutions of the Einstein Equations in dimension three from the point of view of comparison geometry. We find simple monotonic quantities capturing sharply the influence of the Lapse function on the focussing of geodesics. This allows, in particular, a sharp estima…

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converge…

We study Yamabe metrics, and the moduli space of Yamabe metrics, on an arbitrary closed 3-manifold M. The main focus is on the boundary behavior of the moduli space, i.e. the behavior of degenerating sequences of unit volume Yamabe metrics on M. It is proved that such degenerations, when non-trivial in a certain sense,…

New black hole solutions with positive and negative masses in 4 and 5 dimensions.

We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed geometric or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasi-local mass. A variational characterization…

In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold $(M^3,g)$ that admits a smooth nonzero solution $f$ to the equation \begin{align} \label{a1a} \nabla df=ψRc+φg, \end{align} where $ψ,φ$ are given smooth functions of $f$, $Rc$ is the Ricci tensor of $g$. Spaces of this type include various…

We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…

Proves uniqueness of certain spacetime solutions with extremal horizons.

Develops a hedging method for multi-asset derivatives with correlation risk.

New static black hole uniqueness theorems for negative cosmological constant.

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee,…

We prove the existence and the uniqueness of the static dyonic black holes in four dimensional $N=1$ supergravity theory coupled vector and scalar multiplets. We set the near-horizon geometry to be a product of two Einstein surfaces, whereas the asymptotic geometry has to be a space of constant scalar curvature. Using …

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null…

In this article we study self-gravitating static solutions of the Einstein-ScalarField system in arbitrary dimensions. We discuss the existence and the non-existence of geodesically complete solutions depending on the form of the scalar field potential $V(φ)$, and provide full global geometric estimates when the soluti…

Physics-informed methods infer spatial dynamics from static snapshots, but limits exist.