arXiv research

The paper proves boundedness and decay of Teukolsky equations on Kerr backgrounds.

Scattering theory developed for linearised gravity near Schwarzschild black hole.

The paper analyzes Teukolsky equations on Kerr backgrounds, proving boundedness and decay of solutions.

Study on mode stability of gravitational instantons of type D.

Based on operator identities and their formal adjoints, we derive two symmetry operators for the linearized Einstein operator on vacuum backgrounds of Petrov type D and in particular the Kerr spacetime. One of them is of differential order four and coincides with a result of Cohen and Kegeles. The other one is a new op…

The Teukolsky equations are currently the leading approach for analysing stability of linear massless fields propagating in rotating black holes. It has recently been shown that the geometry of these equations can be understood in terms of a connection constructed from the conformal and complex structure of Petrov type…

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…

Kahler geometry explains decoupling of Kerr perturbations.

We prove boundedness and polynomial decay statements for solutions to the spin $\pm2$ generalized Teukolsky system on a Reissner-Nordström background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $…

Scattering theory for linearised gravity on Schwarzschild black hole exterior.

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor $\mathcal{M}_{ab}$ which solves the…

We show that the Teukolsky connection, which defines generalized wave operators governing the behavior of massless fields on Einstein spacetimes of Petrov type D, has its origin in a distinguished conformally and GHP covariant connection on the conformal structure of the spacetime. The conformal class has a (metric com…

The goal of this paper is to provide a geometric framework for analyzing the uniform decay properties of solutions to the Teukolsky equation in the fully nonlinear setting of perturbations of Kerr. It contains the first nonlinear version of the Chandrasekhar transformation introduced in the linearized setting in \cite{…

Study peels tensor equations on Schwarzschild spacetime.

Proves stability of gravitational instantons, proving operator positivity.

New proof of Schwarzschild stability using geometric gauge.

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…

In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner-Nordström family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordström solution arising from regular initial data remain globally bounded…

Study on instability of extreme Reissner-Nordström spacetime perturbations.

Scalar-tensor gravitation theories, such as the Brans-Dicke family of theories, are commonly partly described by a modified Einstein equation in which the Ricci tensor is replaced by the Bakry-Émery-Ricci tensor of a Lorentzian metric and scalar field. In physics this formulation is sometimes referred to as the "Jordan…

Study Galois groupoids of discret Painlevé equations.

Proves solvability of general inverse σ_k equations with constant coefficients.

We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …

Paper establishes estimates for nonlinear equations on compact manifolds.

Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.

The paper generalizes Monge-Ampère equations and their solutions in differential geometry.

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…

Introduces a new PDE involving differential forms for Kähler geometry.

Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescri…

The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.

Paper solves Hessian equations on Kähler manifolds.

The paper introduces new equations in Kähler geometry and proves their solutions and convexity.

We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…

Probabilistic grammars improve equation discovery from data.

In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…

The paper proves constant rank theorems for special Lagrangian equations.

Study solves HJB equations for time-inconsistent control problems.

We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…

In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…

Studies projective geometry and partial differential equations prolongation.

Develops equivariant connections for Yang-Mills equations, simplifying interactions modeling.

The paper derives gradient estimates for solutions of certain equations on metric measure spaces.

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.