Symplectic Reduction and the Homogeneous Complex Monge–Ampère Equation

A Riemannian manifold ( ⁿ , g) is said to be the center of thecomplex manifold ⁿ if is the zero set of a smooth strictly plurisubharmonic exhaustion function ² on such that is plurisubharmonic and solves theMonge–Ampère equation ( ) ⁿ = 0 off , and g is induced by the canonical Kähler metric withfundamental two-form ². Insisting that be unbounded puts severe restrictions on as acomplex manifold as well as on ( , g). It is an open problemto determine the class Riemannian manifolds that are centers of complexmanifolds with unbounded . Before the present work, the list of knownexamples of manifolds in that class was small. In the main result of thispaper we show, by means of the moment map corresponding to isometric actionsand the associated bundle construction, that such class is larger than originally thought and contains many metrically and diffeomorphically`exotic' examples.     

A local regularity of the complex Monge–Ampère equation

We prove a local regularity (and a corresponding a priori estimate) for plurisubharmonic solutions of the nondegenerate complex Monge–Ampère equation assuming that their W ^{2, p} -norm is under control for some p > n(n − 1). This condition is optimal. We use in particular some methods developed by Trudinger and an estimate for the complex Monge–Ampère equation due to Kołodziej.     

Dirichlet problem of quaternionic Monge–Ampère equations

In this paper, the author studies quaternionic Monge–Ampère equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper aims to answer the question proposed by Semyon Alesker in [3]. It also extends relevant results in [8] to the quaternionic vector space.     

Quasi-Neutral Limit of the Euler–Poisson and Euler–Monge–Ampère Systems

ABSTRACT

This paper studies the pressureless Euler–Poisson system and its fully nonlinear counterpart, the Euler–Monge–Ampère system, where the fully nonlinear Monge–Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.     

Plurifinely Plurisubharmonic Functions and the Monge Ampère Operator

We will define the Monge-Ampère operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets U???? ⁿ and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that in the Dirichlet norm weakly plurifinely plurisubharmonic functions are locally limits of plurisubharmonic functions. As a consequence of the latter, weakly plurifinely plurisubharmonic functions are strongly plurifinely plurisubharmonic outside of a pluripolar set.     

On the Dirichlet Problem for a Class of Singular Complex Monge–Ampère Equations

We study the Dirichlet problem of the n-dimensional complex Monge–Ampère equation det(uij) = F/|z|~(2α), where 0 α n. This equation comes from La Nave–Tian's continuity approach to the Analytic Minimal Model Program.     

Monge–Ampère foliations for degenerate solutions

We study the problem of the existence and the holomorphicity of the Monge–Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge–Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a positive answer to a question of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge–Ampère equation and we obtain a generalization the work of Wong on the classification of complete weighted circular domains.     

On the Monge–Ampère equation with boundary blow-up: existence,uniqueness and asymptotics

We consider the Monge–Ampère equation det D ² u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ?Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) is positive in Ω and non-negative on ?Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in \({\mathbb R}^N\) with N ≥ 2. We give asymptotic estimates of the behaviour of such solutions near ?Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ?Ω and \(b\equiv 0\) on ?Ω.     

W 2,1 regularity for solutions of the Monge–Ampère equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, is $W^{2,1}_{\mathrm{loc}}$ . This is obtained by showing higher integrability a priori estimates for D ² u, namely D ² u∈Llog ^k L for any k∈?.     

Monge–Ampère measures on pluripolar sets

In this article we solve the complex Monge–Ampère problem for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kołodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge–Ampère measure, then it is a complex Monge–Ampère measure.     

On global solvability of initial value problem for hyperbolic Monge–Ampère equations and systems

The communication concerns a theory of global solvability of initial value problem for nonlinear hyperbolic equations with two independent variables that is an immediate analog of a theory of global solvability of ordinary differential equations.     

The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space

In this paper, we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic Monge–Ampère operator. By introducing a new quaternionic capacity, we prove a sufficient condition which implies the weak convergence of quaternionic Monge–Ampère measures \((\triangle u_j)^n\rightarrow (\triangle u)^n\). We also obtain an equivalent condition of “convergence in \(C_{n-1}\)-capacity” by using methods from Xing (Proc Am Math Soc 124(2):457–467, 1996). As an application, the range of the quaternionic Monge–Ampère operator is discussed.     

A meshfree method for solving the Monge–Ampère equation

Numerical Algorithms - This paper solves the two-dimensional Dirichlet problem for the Monge-Ampère equation by a strong meshless collocation technique that uses a polynomial trial space and...     

Regularity of the Monge–Ampère equation in Besov’s spaces

Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi $ be the optimal transportation mapping pushing forward $\mu $ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$ . In this paper, we introduce a new approach to the regularity problem for the corresponding Monge–Ampère equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi )}$ in the Besov spaces $W^{\gamma ,1}_{loc}$ . We prove that $D^2 \Phi \in W^{\gamma ,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 \Phi \in L^p_{loc}$ for some $p>1$ . Our proof does not rely on the previously known regularity results.