Singular Monge-Ampere foliations

Hölder continuous solutions of the Monge-Ampère equation on compact Hermitian manifolds

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.

Monge-Ampere 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 P.M. Wong on the classification of complete weighted circular domains.

The complex Monge-Ampère equation on compact Hermitian manifolds

We show that, up to scaling, the complex Monge-Ampere equation on compact Hermitian manifolds always admits a smooth solution.

Weak solutions to Monge-Ampère type equations on compact Hermitian manifold with boundary

We prove the bounded subsolution theorem for the complex Monge-Ampère type equation, with the right hand side being a positive Radon measure, on a compact Hermitian manifold with boundary.

Finite Type Monge-Ampère Foliations

In this paper we extend our previous work on singularities of Monge-Ampère foliations to the case of pseudoconvex finite type domains. We are able to answer the questin of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge-Ampère equation completely in dimension 2 . We are also able to generalize the work of P.M. Wong in dimension 2 on the classification of complete weighted circular domains to include finite type domains.

Regularity of geodesic rays and Monge-Ampere equations

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,α}, 0<α<1$. An essential step is to establish that the rays can be extended as solutions of a Dirichlet problem for a Monge-Ampere equation on a Kaehler manifold which is compact.

Complex Monge-Ampere equations on Hermitian manifolds

We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an application we generalize existing results on the Donaldson conjecture on geodesics in the space of Kahler metrics to the Hermitian setting.

Monge-Ampere boundary measures

We study swept-out Monge-Ampere measures of plurisubharmonic functions and boundary values related to these measures.

The $L^{\infty}$ estimates for parabolic complex Monge-Ampere and Hessian equations

In this paper, we consider a version of parabolic complex Monge-Ampere equations, and use a PDE approach similar to Phong et al to establish $L^{\infty}$ and Hölder estimates. We also generalize the $L^{\infty}$ estimates to parabolic Hessian equations.

A Decomposition of Complex Monge-Ampere Measures

We prove one decomposition theorem of complex Monge-Ampere measures of plurisubharmonic functions in connection with their pluripolar sets.

Gradient estimate for complex Monge-Ampere equation with continuous right hand side

In this note, we consider complex Monge-Ampere equation posed on a compact Kähler manifold. We show how to get $L^p$($p<\infty$) and $L^{\infty}$ estimate for the gradient of the solution in terms of the continuity of the right hand side.

Complex Monge-Ampere of a Maximum

We give a formula for the complex Monge-Ampere operator applied to the maximum of a finite number of functions.

A Monge-Ampere Type Fully Nonlinear Equation on Hermitian Manifolds

We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.

Choquet-Monge-Ampere Classes

We introduce and study Choquet-Monge-Ampere classes on compact Kahler manifolds. They consist of quasi-plurisubharmonic functions whose sublevel sets have small enough asymptotic Monge-Ampere capacity. We compare them with finite energy classes, which have recently played an important role in Kahler Geometry.

Some properties of the complex Monge-Ampere operator in Cegrell's classes and applications

In this article we will first prove a result about convergence in capacity. Using the achieved result we will obtain a general decompositon theorem for complex Monge-Ampere measues which will be used to prove a comparison principle for the complex Monge-Ampere operator.

Locally Monge-Ampere Foliations

It is shown that codimension one parabolic foliations of complex manifolds are holomorphic. This is proved using the fact that codimension one foliations of complex manifolds are necessarily locally Monge-Ampère foliations and that parabolic leaves cannot have hyperbolic behavior. The result holds true also for locally Monge-Ampère foliations with parabolic leaves of arbitrary codimension.

Monge-Ampere measures on pluripolar sets

In this article we solve the complex Monge-Ampere equation 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 Kolodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampere measure, then it is a complex Monge-Ampere measure.

Stability of Bounded Solutions for Degenerate Complex Monge-Ampère equations

We generalize and strenghten Kołodziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Ampère equation.

The $\mathcal C^{2,α}$ estimate of complex Monge-Ampere equation

We prove that any $\mathcal C^{1,1}$ solution to complex Monge-Ampère equation $det(u_{i\bar{j}})=f$ with $0<f\in\mathcal C^α$ is in $\mathcal C^{2,α}$ for $α\in (0,1)$.

Cegrell classes and a variational approach for the quaternionic Monge-Ampere equation

In this paper, we introduce finite energy classes of quaternionic plurisubharmonic functions of Cegrell type and study the quaternionic Monge-Ampere operator on these classes on quaternionic hyperconvex domains of Hn. We extend the domain of definition of quaternionic Monge-Ampere operator to some Cegrell classes, the functions of which are not necessarily bounded. We show that integration by parts and comparison principle are valid on some classes. Moreover, we use the variational method to solve the quaternionic Monge-Ampere equations when the right hand side is a positive mea- sure of finit…

Monge-Ampere measures on subvarieties

In this article we address the question whether the complex Monge-Ampère equation is solvable for measures with large singular part. We prove that under some conditions there are no solution when the right-hand side is carried by a smooth subvariety in $\CEP{n}$ of dimension $k<n$.

Log canonical thresholds and Monge-Ampere masses

In this paper, we prove an inequality for log canonical thresholds and Monge-Ampere masses. The idea of proof is a combination of the Ohsawa-Takegoshi $L^2$-extension theorem and inequalities in [ACKHZ09], [DH14]. It also give an analytic proof for the main result in [DH14] in the case of dimension 2.

Hermitian流形上复Monge-Ampère方程解的连续性模

Modulus of continuity for solutions to complex Monge-Ampère equations on Hermitian manifolds

本文证明了紧Hermitian流形上复Monge-Ampère方程解的一致对数连续性，推广了Kähler情形的已知结果。

Subsolution theorem and the Dirichlet problem for the quaternionic Monge-Ampere equation

In this paper, the author studies quaternionic Monge-Ampère equations and obtain the existence of the solutions to the Dirichlet problem for such equations in strictly pesudoconvex domains in quaternionic space. The stability and subsolution theorems are established for quaternionic Monge-Ampère equations.

Parabolic complex Monge-Ampere equations on compact Kahler manifolds

We study the long-time existence and convergence of general parabolic complex Monge-Ampere type equations whose second order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution.

具有预定奇点的 Monge-Ampère 势的连续性

Continuity of Monge-Ampère Potentials with Prescribed Singularities

本文研究带预定奇点的复 Monge-Ampère 方程解的连续性，推广了已有结果，并据此实现从预定奇点流启动 Monge-Ampère 流。

Complex Monge-Ampere equations with solutions in finite energy classes

We characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Ampère equation has a solution of finite pluricomplex energy. Our results are also valid in the big cohomology class setting.

Families of Monge-Ampere measures with Holder continuous potentials

Let X be a compact Kahler manifold. Let F be a family of probability measures on X whose superpotentials are Holder continuous with uniform Holder exponent and Holder constant. We prove that the corresponding family of solutions of the complex Monge-Ampere equations whose right-hand sides are measures in F is Holder continuous.

An analytic proof of the Krylov estimates for the complex Monge-Ampere equation and applications

We provide an analytic proof of a theorem of Krylov dealing with global $C^{1,1}$ estimates to solutons of degenerate complex Monge-Ampère equations. As an application we show optimal regularity for various extremal functions with nonconstant boundary values.

The Non-Archimedean Monge-Ampere Equation

We give an introduction to our work on the solution to the non-Archimedean Monge-Ampere equation and make comparisons to the complex counterpart. These notes are partially based on talks at the 2015 Simons Symposium on Tropical and Nonarchimedean Geometry.