arXiv research

We study a generalized Abreu equation and derive some estimates.

We consider a fourth order partial differential equation in n-dimensional space introduced by Abreu in the context of Kähler metrics on toric orbifolds. Similarity solutions depending only on the radial coordinate in R^n are determined in terms of a second order ordinary differential equation. A local asymptotic analys…

We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.

In this paper we prove the interior regularity for the solution to the Abreu equation in any dimension assuming the existence of the $C^0$ estimate.

We study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.

We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-We…

In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.

The paper develops various estimates for solutions of a fourth order nonlinear PDE, which corresponds to prescribing the scalar curvature of a toric Kahler metric.

We study the Dirichlet problem of the Abreu equation. The solutions provide the Kahler metrics of constant scalar curvature on the complex torus.

This paper consists of real and complex affine techniques for studying the Abreu equation on toric surfaces. In particular, an interior estimate for Ricci tensor is given.

The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions.

In this paper, we study the generalized Abreu equation on a Delzant ploytope $Δ\subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.

New complete hypersurfaces found in affine geometry.

Extends boundary estimates for Monge-Ampère equations in polygonal domains.

The study finds conditions for the existence of extremal toric almost Kähler metrics.

This note is a step towards demonstrating the benefits of a symplectic approach to studying equivariant Kähler geometry. We apply a local differential geometric framework from Kähler toric geometry due to Guillemin & Abreu to the case of the standard linear $\SU(n)$ action on $\bbC^n\setminus\{0\}$. Using this framewor…

Motivated by the work of Abreu and Freitas, we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over $\mathbb{p}^1$.

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a we…

New system modifies constant scalar curvature Kähler condition with a 'Higgs field'.

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donalds…

We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the s…

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

In this paper we study the topology of the space $\I_ω$ of complex structures compatible with a fixed symplectic form $ω$, using the framework of Donaldson. By comparing our analysis of the space $\I_ω$ with results of McDuff on the space $\cat J_ω$ of compatible almost complex structures on rational ruled surfaces, we…

In this paper, we show that the complete scalar-flat Kahler metrics constructed by Abreu and the author on strictly unbounded toric 4-dimensional orbifolds have finite $L^2$ norm of the full Riemannian tensor. In particular, this answers a question of Donaldon's on the corresponding Generalized Taub-NUT metric on $R^4$…

Researchers found unique scalar-flat Kähler metrics on toric surfaces.

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metri…

Curvature on Kähler toric manifolds

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…

In this paper we study the smallest non-zero eigenvalue $λ_1$ of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for $λ_1$ in terms of moment polytope data. We show that this bound can only be attained for $\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\mathbb{CP}^n$ endowe…

We prove that the quasi-Einstein metrics found by Lü, Page and Pope on $\mathbb{C}P^{1}$-bundles over Fano Kähler-Einstein bases are conformally Kähler and that the Kähler class of the conformal metric is a multiple of the first Chern class. A detailed study of the lowest-dimensional example of such metrics on $\mathbb…

Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric Kähler metric g. Abreu asked whether the spectrum of the Laplace operator $Δ_g$ on $\mathcal{C}^\infty(M)$ determines the moment polytope of M, and hence by Delzant's theorem determines M up to symplectomorphism. We report on some progre…

We unify and extend a number of approaches related to constructing multivariate Variance-Gamma (V.G.) models for option pricing. An overarching model is derived by subordinating multivariate Brownian motion to a subordinator from the Thorin (1977) class of generalised Gamma convolution subordinators. A class of models …

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere $S^2$ endowed with $S^1$-invariant metrics, we consider the subsequence $λ_k^G$ of the spectrum of a Riemannian manifold $M$ which corresponds…

We pursue the symplectic description of toric Kahler manifolds. There exists a general local classification of metrics on toric Kahler manifolds equipped with Hamiltonian two-forms due to Apostolov, Calderbank and Gauduchon(ACG). We derive the symplectic potential for these metrics. Using a method due to Abreu, we rela…

We study the symplectomorphism groups $G_λ=Symp_0(M,ω_λ)$ of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms $ω_λ$ with variable cohomology class. We show that the existence of nontrivial elements in $π_*({\cal A},{\cal A}')$, where $({\cal A},{\cal A}')$ is a suitable pair of spac…

Given a compact symplectic toric manifold $(M,ω, \mathbb{T})$, we identify a class $DGK_ω^{\mathbb{T}}(M)$ of $\mathbb{T}$-invariant generalized Kähler structures for which a generalisation the Abreu-Guillemin theory of toric Kähler metrics holds. Specifically, elements of $DGK_ω^{\mathbb{T}}(M)$ are characterized by t…

Extends Kähler metrics theory to symplectic manifolds with toric actions.

Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different …

Paper finds new equations for pseudospherical surfaces with isometric immersions.

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.