We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kaehler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kaehler metrics and Berndtsson's positivity of direct images we extend Ding-Tian's variational characterization and Bando-Mabuchi's uniqueness result to singular Kaehler-Einstein metrics. Finally using our variational characterization we prove the existence, uniqueness and convergence of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy in our sense.
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
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.
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.
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.
Let $(X,ω)$ be a compact Kähler manifold. We prove that all Monge-Ampère capacities are comparable. Using this we give an alternative direct proof of the integration by parts formula for non-pluripolar products recently proved by M. Xia.
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.
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.
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.
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.
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.
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…
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.
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.
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.
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.
We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.
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.
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.
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.
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.
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.
In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Ampère equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates for fully non-linear equations in Kähler geometry. As an application, we derive a uniform diameter bound for Kähler metrics satisfying certain Monge-Ampère equations.
These are the lecture notes for the Morningside Center of Mathematics Geometry Summer School on August 15-20, 2022. These lectures sketch the results by Yau, Demailly-Paun, the author, and Datar-Pingali about generalized Monge-Ampère equations and subvarieties and aim to use these results to study the Hodge conjecture.
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.
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.
We prove the Hölder continuity of the unique solution to quaternionic Monge-Ampère equation with densities in $L^{p},$ $p>2,$ on a bounded strictly pseudoconvex domains.
The main goal of this article is to find, following the approach given in [Ce1] and [Ce2], the largest possible sub-class of plurisubharmornic functions on a complex variety on which the complex Monge-Ampère operator can be reasonably defined. Moreover, we are also interested in giving sufficient condition for solvability of the Monge-Ampère operator Monge-Ampère operators on complex varieties.
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.
In this paper, we prove the Hölder continuity for solutions to the complex Monge-Ampère equations on non-smooth pseudoconvex domains of plurisubharmonic type ${m}$.
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.
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.
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.
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.
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.
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.
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.
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.
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 $α\in (0,1)$.
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.
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…
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.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equations.
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 $k<n$.
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".
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.
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.
Let $μ^z$ be the measure obtained by sweeping out the Monge-Ampère measure of the pluricomplex Green function with pole at $z. $ We prove that $μ^z$ vanish on Levi flat parts of the boundary for 1) every relatively compact analytic polyhedron in complex space, 2) product domains of hyperconvex sets in Stein manifolds.
