Complex Monge-Ampere of a Maximum

Hölder continuous solutions to the complex Monge-Ampère equations in non-smooth pseudoconvex domains

In this paper, we prove the Hölder continuity for solutions to the complex Monge-Ampère equations on non-smooth pseudoconvex domains of plurisubharmonic type ${m}$.

An Inequality for Mixed Monge-Ampère measures

We generalize an inequality for mixed Monge-Ampère measures. We also give an example that shows that our assumptions are sharp. The corresponding result in the setting of compact Kähler manifold is also discussed.

On a Monge-Ampère operator for plurisubharmonic functions with analytic singularities

We study continuity properties of generalized Monge-Ampère operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a formula for the total mass of the Monge-Ampère measure of such a function on a compact Kähler manifold.

Weak convergence of Monge-Ampère measures on compact Hermitian manifolds

We give a sufficient condition on a sequence of uniformly bounded $ω$-plurisubharmonic functions, $ω$ being a Hermitian metric, for which the sequence of associated Monge-Ampère measures converges weakly. This criterion can be used to obtained a bounded $ω$-plurisubharmonic solution to the Monge-Ampère equation.

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.

Local Hölder continuity of solutions of the complex Monge-Ampère equation

In this paper, we are interested in studying the Dirichlet problem for the complex Monge-Ampère operator. We provide necessary and sufficient conditions for the problem to have Hölder continuous solutions.

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.

Complex Monge-Ampère equations for plurifinely plurisubharmonic functions

This paper studies the complex Monge-Ampère equations for $\mathcal F$-plurisubharmonic functions in bounded $\mathcal F$-hyperconvex domains. We give sufficient conditions for this equation to solve for measures with a singular part.

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.

A local regularity for the complex Monge-Ampère equation

We prove a local regularity (and a corresponding a priori estmate) for plurisubharmonic solutions of the nondegenerate complex Monge-Ampére 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 $L^q$-estimate for the complex Monge-Ampére equation due to Kołodziej.

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 $α\in (0,1)$.

On the Hölder continuous subsolution problem for the complex Monge-Ampère equation

We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge-Ampère equation admits Hölder continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge-Ampère total mass, we obtain an affirmative answer to a question of Zeriahi.

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 $k<n$.

Complex Monge-Ampère equation in strictly pseudoconvex domains

We study the complex Monge-Ampère equation $(dd^c u)^n=μ$ in a strictly pseudoconvex domain $Ω$ with the boundary condition $u=φ$, where $φ\in C(\partialΩ)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside "small sets".

On the Maximal Rank Problem for the Complex Homogeneous Monge-Ampère Equation

We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Ampère Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank.

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.

The quaternionic Monge-Ampère operator and plurisubharmonic functions on the Heisenberg group

Many fundamental results of pluripotential theory on the quaternionic space $\mathbb{H}^n$ are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge-Ampère operator, differential operators $d_0$ and $d_1$ and a closed positive current on the Heisenberg group. The quaternionic Monge-Ampère operator is the coefficient of $ (d_0d_1u)^n$. We establish the Chern-Levine-Nirenberg type estimate, the existence of quaternionic Monge-Ampère measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternioni…

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。

复Monge-阿普雷运算符在紧赫尔米特流形上的连续性

Continuity of the complex Monge-Ampère operator on compact Hermitian manifolds

本文研究了复Monge-阿普雷运算符在紧赫尔米特流形上的连续性与弱收敛性，提出了若干结论，并在存在光滑子解的假设下，找到了复Monge-阿普雷方程的弱解。

Weak subsolutions to complex Monge-Ampère equations

We compare various notions of weak subsolutions to degenerate complex Monge-Ampère equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge-Ampère inequalities due to Kolodziej and Dinew.

On the invariance of the total Monge-Ampere volume of Hermitian metrics

In this note, we describe the Hermitian metrics that leave the total Monge-Ampere volume invariant. In particular, we give several characterizations of the Hermitian metrics which satisfy the comparison principle for the complex Monge-Ampere operator

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.

Weak Solutions To Complex Monge-Ampère Equation on hyperconvex domains

We show a very general existence theorem to the complex Monge-Ampère type equation on hyperconvex domains.

Continuity of the Complex Monge-Ampere Operator on Compact Kahler Manifolds

We prove several approximation theorems of the complex Monge-Ampere operator on a compact Kahler manifold. As an application we give a new proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex Monge-Ampere operator in the class of w-plurisubharmonic functions with vanishing complex Monge-Ampere mass on all pluripolar sets. As a by-product we obtain a stability theorem of solutions of complex Monge-Ampere equations in some subclass.

Dirichlet problem of complex Monge-Ampère equation near an isolated KLT singularity

We solve the Dirichlet Problem of Monge-Ampère equation near an isolate Klt singularity, which generalizes the result of Eyssidieux-Guedj-Zeriahi \cite{EGZ}, where the Monge-Ampère equation is solved on singular varieties without boundary. As a corollary, we construct solutions to Monge-Ampère equation with isolated singularity on strongly pseudoconvex domain $Ω$ contained in $\mathbb{C}^n$.

Weak Solutions to Complex Monge-Ampère Equations on Compact Kähler Manifold

We show a general existence theorem to the complex Monge-Ampère type equation on compact Kähler manifolds.

Envelope Approach to Degenerate Complex Monge-Ampère Equations on compact Kähler manifold

We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

The Monge-Ampère equation on almost complex manifolds

We study the Dirichlet problem for the Monge-Ampère equation on almost complex manifolds. We obtain the existence of the unique smooth solution of this problem in strictly pseudoconvex domains.

On Degenerated Monge-Ampere Equations over Closed Kähler Manifolds

In this work, we study Monge-Ampere equations over closed Kähler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.

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 degenerate complex Monge-Ampére flows I

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the first of a series on this subject, is to develop a viscosity theory for degenerate complex Monge-Ampère flows in domains of $\C^n$.

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.

Hölder continuity of solutions to the complex Monge-Ampère equation with the right hand side in L^p. The case of compact Kähler manifolds

We prove that on compact Kähler manifolds solutions to the complex Monge-Ampère equation, with the the right hand side in $L^p, p>1,$ are Hölder continuous.

Singular Monge-Ampere foliations

This replaces the previous version, by correcting an error in the proof of Theorem 1.4, that was pointed out by the referee.

Parabolic complex Monge-Ampère equations on compact Hermitian manifolds

We prove the long-time existence and convergence of solutions to a general class of parabolic equations, not necessarily concave in the Hessian of the unknown function, on a compact Hermitian manifold. The limiting function is identified as the solution of an elliptic complex Monge-Ampère equation.

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.

Domains of definition of Monge-Ampère operators on compact Kähler manifolds

Let $(X,ω)$ be a compact Kähler manifold. We introduce and study the largest set $DMA(X,ω)$ of $ω$-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all $\om$-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all $ω$-psh functions. As a consequence, any $\om$-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy.

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.

The weigthed Monge-Ampère energy of quasiplurisubharmonic functions

We study degenerate complex Monge-Ampère equations on a compact Kähler manifold $(X,ω)$. We show that the complex Monge-Ampère operator $(ω+ dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,ω)$ of $ω$-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class ${\mathcal E}(X,ω)$ is the largest class of $ω$-psh functions on which the Monge-Ampère operator is well-defined and the comparison principle is valid. It contains several functions whose gradient is not square integrable. We give a complete description of the range of the Monge-Ampère operator $(ω+dd^c \cdo…

Complex Monge-Ampère equations on quasi-projective varieties

We introduce generalized Monge-Ampère capacities and use these to study complex Monge-Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor.

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.

Weak solutions to the complex Monge-Ampère equation on Hermitian manifolds

The main result asserts the existence of continuous solutions of the complex Monge-Ampère equation with the right hand side in $L^p, p>1$, on compact Hermitian manifolds.

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.

Regularity of A Complex Monge-Ampère Equation on Hermitian Manifolds

We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hatω$-plurisubharmonic solution of an elliptic complex Monge-Ampère equation is smooth under an assumption on the background Hermitian metric $\hatω$. This generalizes a result of Székelyhidi and Tosatti on Kähler manifolds.

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.

复蒙日-安培特征值问题的一种迭代方法

An Iterative Approach to the Complex Monge-Ampère Eigenvalue Problem

本文针对有界严格拟凸域上的狄利克雷复蒙日-安培特征值问题，借鉴实蒙日-安培算子的已有迭代思路，提出一种新的迭代逼近方法。

Stability of solutions to complex Monge-Ampère equations in big cohomology classes

We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known to hold in the context of Kähler classes.

Regularizing properties of Complex Monge-Ampère flows

We study the regularizing properties of complex Monge-Ampère flows on a Kähler manifold $(X,ω)$ when the initial data are $ω$-psh functions with zero Lelong number at all points. We prove that the general Monge-Ampère flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solution.