arXiv research

The paper studies sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.

We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact Kähler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on Kähler groups proved using the Donaldson-Corlette-Hitchin-Simpson corresponde…

In this article, we study the Kapustin-Witten equations on a closed, simply-connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use the Taubes' compactness theorem in arXiv:1307.6447v4 to prove that if $(A,φ)$ is a smooth solution of Kapustin-Witten equations and the connection $A$ is …

The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.

Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.

We define a functional ${\cal J}(h)$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles \cite{Kobayashi}, and study some of its basic properties. We show that ${\c…

The Kapustin-Witten equations relate to nonabelian Hodge theory on Kähler surfaces.

We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…

In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in SL(2,R) with prescribed behavior at the punctures and nonzero Euler number, following the strategy employed by Hitchin in the unpunctured case and exploiting Hitchin-Simpson correspon…

We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…

This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship to representations of the fundamental group of X. We construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces a…

We classify the connected components of the space of representations of the fundamental group of a closed oriented surface of genus $\geq 2$ in $Sp(4,{\mathbf R})$. We prove that this is equivalent to classifying the connected components of the moduli space of representations (which consists of conjugacy classes of red…

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the ba…

The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold $(X, ω)$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend thi…

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.

Study a modified Laplacian equation in spacetime.

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

The paper proves estimates for vortex-type equations on compact Riemann surfaces.

Geometrically interprets two equations, showing their equivalence and providing solutions.

Auxiliary equations improve bounds in symplectic geometry.

In this paper we introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi-Yau metric. New equation differs from the Monge--Ampere equation considered by Calabi and Yau.