相关论文: Exterior Monge-Ampere Solutions
We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…
We study the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are H\"older…
We study graphs of positive extrinsic curvature with a non-removable isolated singularity in 3-dimensional warped product spaces, and describe their behavior at the singularity in several natural situations. We use Monge-Amp\`ere equations…
We give sufficient conditions for the existence and uniqueness, in bounded uniformly convex domains $\Omega$, of solutions of degenerate elliptic equations depending also on the nonlinear gradient term $H$, in term of the size of $\Omega$,…
In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Amp\`ere equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal…
Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is…
We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santal\'o point is at the origin, with those of its inner and outer regularizations. We also provide functional…
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian…
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the…
In this paper, we are concerned with the monotonic and symmetric properties of convex solutions Monge-Amp\`ere systems for instance, considering \begin{equation*} \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m, \end{equation*} over…
We obtain the existence, uniqueness and regularity results for solutions to kinetic Fokker-Planck equations with bounded measurable coefficients in the presence of boundary conditions, including the inflow, diffuse reflection and specular…
Regularity properties of intrinsic objects for a large class of Stein Manifolds, namely of Monge-Amp\`ere exhaustions and Kobayashi distance, is interpreted in terms of modular data. The results lead to a construction of an infinite…
We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on…
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally…
In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Amp\`ere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial…
Surfaces of constant negative curvature in Euclidean space can be described by either the sine-Gordon equation for the angle between asymptotic directions, or a Monge-Ampere equation for the graph of the surface. We present the explicit…
Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem…
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic…
We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for…
This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and…