arXiv research

Unique solutions found for wave-like decaying null infinity equations.

Unique global solutions found for specific initial data.

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

In this paper we exploit the ideas and formalisms of twistor theory, to show how, on Minkowski space, given a null solution of the wave equation, there are precisely two null directions in $\ker df$, at least one of which is a shear-free ray congruence.

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partialΩ_0$, we show that there exists a weak solution to the null mean curvatu…

The paper explores geometric invariants of null hypersurfaces using Carrollian geometry.

Integrable flows on null curves in anti-de Sitter 3-space studied.

It is of interest to study supergravity solutions preserving a non-minimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the spacetime admits a Killing spinor and hence a null or timelike Killing vector field. Any spacetime admitting a covariantly constant null vector fiel…

Proves global existence and uniqueness of solutions for Einstein-scalar-field equations.

In this paper, we carry out null space analysis for Class-Specific Discriminant Analysis (CSDA) and formulate a number of solutions based on the analysis. We analyze both theoretically and experimentally the significance of each algorithmic step. The innate subspace dimensionality resulting from the proposed solutions …

Geometric transformations on null curves in AdS induce KdV solutions.

The study connects electromagnetic structures to Legendrian fields on the 3-sphere.

We present a definition of null G-structures on Lorentzian manifolds and investigate their geometric properties. This definition includes the Robinson structure on 4-dimensional black holes as well as the null structures that appear in all supersymmetric solutions of supergravity theories. We also identify the induced …

We get decay rate of higher derivatives of nonlinear massless Dirac equations with a kind of "good" spin null form. The method we rely on is similar to that of Li and Zang. However, they only give the decay rate of solution itself to nonlinear massless Dirac system.

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the traje…

Novel approach to wave equations near null infinity in flat spacetimes.

We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions…

Study spin-0 fields on n-dimensional Minkowski spacetimes, computing asymptotic charges.

The paper addresses fairness in machine learning models through structural econometrics, projecting indexes into null spaces to find fair solutions.

Study reveals hidden null components in overparametrized neural networks.

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

Null Kähler metrics are characterized by Painlevé I or II ODEs.

New methods for constructing null fluid metrics and solving optical lift conjectures.

Study of Randers spacetimes and their Finsler gravity solutions.

New contact structures extend supergravity solutions.

We study the nonlinear stability of the $(3+1)$-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linea…

We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, …

The Kastor-Traschen metric is a time-dependent solution of the Einstein-Maxwell equations with positive cosmological constant $Λ$ which can be used to describe an arbitrary number of charged dynamical black holes. In this paper, we consider the null geodesic structure of this solution, in particular, focusing on the pr…

We introduce a collection of 1/2-$π_1$-null 4-dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the $π_1$-null kernels which are known to admit a solution in the topological category. Using geometric applications of the group-theoretic 2-Engel rela…

The paper studies hyperbolic equations in a spacetime foliation, proving existence and uniqueness.

We find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group O(n) in terms of free holomorphic data which lead to formulae for all such harmonic maps. Our work reveals an interesting correspondence between certain harmonic maps and the free Wei…

We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting…

Study shows instability of naked singularities in scalar field models.

Survey on geometric, analytic, and topological aspects of 4D equations.

The paper extends isometric embedding results to null cones and spheres.

Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.

We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs e…

We give a local characterization of codimension two submanifolds which are marginally trapped in Robertson-Walker spaces, in terms of an algebraic equation to be satisfied by the height function. We prove the existence of a large number of local solutions. We refine the description in the case of curves with null accel…

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…

We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of sym…

In this paper we prove that every bordered Riemann surface M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean $3$-space $\mathbb{C}^3$. The real part of such an embedding is a complete conformal minimal immersion $M\to \mathbb{R}^3$ with bounded image. For any such $M$ we also co…

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…

Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.

Proves existence of solutions with concentrated energy in 2+1 spacetime.

Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.

Study null geodesics on even-dimensional conformal manifolds, finding Einstein metrics and CR structures.

The kernel null-space technique and its regression-based formulation (called one-class kernel spectral regression, a.k.a. OC-KSR) is known to be an effective and computationally attractive one-class classification framework. Despite its outstanding performance, the applicability of kernel null-space method is limited d…

Study proves global existence and decay for complex wave equations.