arXiv research

New methods using spacetime harmonic functions solve geometric inequalities.

The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.

Study harmonicity of metrics in generalized Kantowski-Sachs spacetime.

Paper bounds mass of 3D Einstein data using spacetime harmonic functions.

New proof removes decay assumptions for spacetime positive mass theorem.

Defines a new quasi-local mass related to spacetime harmonic functions.

The paper proves energy theorems for specific initial data sets in 3D spacetime.

Initial data with zero mass must be in pp-wave spacetimes.

The paper studies a new vacuum field equation and its solutions.

Static spacetimes are stable attractors in a flow equation.

Estimates bandwidth for CMC initial data sets.

Theorem proves spectral rigidity of warped product metrics.

Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.

Rigidity theorem shows massless hyperboloidal data embeds into Minkowski space.

Proves density and mass theorems for specific initial data sets.

Study rigidifies geometry of electrostatic systems with specific tensor properties.

New proof of stability for expanding Kerr-de Sitter spacetimes with smoothness at the boundary.

In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the g…

The paper introduces comprehensive quasi-Einstein spacetimes and explores their properties.

Solves initial boundary value problem for vacuum Einstein equations and proves geometric uniqueness.

We prove boundedness and polynomial decay statements for solutions to the spin $\pm1$ Teukolsky-type equation projected to the $\ell=1$ spherical harmonic on Reissner-Nordström spacetime. The equation is verified by a gauge-invariant quantity which we identify and which involves the electromagnetic and curvature tensor…

In this paper, we study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d'Alembertian equation. In particular, we prove the solution decays at rate $τ^{-1+δ}$ t…

Proves existence and uniqueness of vacuum black hole solutions in higher dimensions.

Researchers found a way to measure energy in black hole perturbations.

We characterize $η$-Ricci solitons $(g,ξ,λ,μ)$ in some special cases when the $1$-form $η$, which is the $g$-dual of $ξ$, is a harmonic or a Schrödinger-Ricci harmonic form. We also provide necessary and sufficient conditions for $η$ to be a solution of the Schrödinger-Ricci equation and point out the relation between …

Establishes inequality for multiple black holes, proving mass lower bound.

Paper proves stronger Penrose inequality with matter density.

The study characterizes spacetime and modified gravity models using projective curvature tensor.

The paper explores geometric relationships in manifolds with curvature constraints, proving new inequalities and rigidity results.

Researchers prove a Penrose inequality for spacetime with specific conditions.

This work proves Kerr black holes are dynamically stable under certain perturbations.

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the ex…

The integral of the energy density function $\mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $Λ$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this …

Paper introduces a new time separation function for $C^0$ spacetimes.

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 energy-momentum vector $(E,P)$ of the initial data satisfies the inequality $E \ge |P|$. Previously, this theorem was proven…

We solve Bartnik's stationary extension problem near Schwarzschild spheres.

Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{μν})$ or an initial data set $(Σ, h_{ij}, K_{ij})$ admitting a suitably defined convex function. We show how…

Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{μν})$ or an initial data set $(Σ, h_{ij}, K_{ij})$ admitting a suitably defined convex function. We show how…

We prove that every $(3+1)$-dimensional flat GHMC Minkowski spacetime which is not a translation spacetime or a Misner spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In otherwords, we prove that every such spacetime carries a unique time function with isochrones of constan…

For a stable marginally outer trapped surface (MOTS) in an axially symmetric spacetime with cosmological constant $Λ> 0$ and with matter satisfying the dominant energy condition, we prove that the area $A$ and the angular momentum $J$ satisfy the inequality $8π|J| \le A\sqrt{(1-ΛA/4π)(1-ΛA/12π)}$ which is saturated pre…

In 1966, Jenkins and Serrin gave existence and uniqueness results for infinite boundary value problems of minimal surfaces in the Euclidean space, and after that such solutions have been studied by using the univalent harmonic mapping theory. In this paper, we show that there exists a one-to-one correspondence between …

This paper continues the investigation of constant mean curvature (CMC) time functions in maximal globally hyperbolic spatially compact spacetimes of constant sectional curvature, which was started in math.DG/0604486. In that paper, the case of flat spacetimes was considered, and in the present paper, the remaining cas…

Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.

We develop a new approach to the existence of time functions on Lorentzian manifolds, based on Conley's work regarding Lyapunov functions for dynamical systems. We recover Hawking's result that a stably causal admits a time function through a more general result giving the existence of a continuous function that is non…

Study examines maximal domains of radial harmonic functions across different curvature types.

The paper investigates geometrical aspects of static spacetime with almost gradient Ricci solitons.

Study of spacetime dynamics in 2+1 gravity leads to Thurston boundary.

Exploring distance functions on spacetime models.