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.
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.
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.
Let M be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of $TM$. We prove here that the only real analytic Zoll metric on the $n$-sphere with an entire adapted complex structure on $TM$ is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in ter…
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 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.
By studying a complex Monge-Ampère equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kähler manifold $N^n$ with $\Ric_k< 0$ for some integer $k$ with $1<k<n$, and the ampleness of the canonical line bundle $K_N$.
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.
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.
We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.
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.
The proof of the comparison principlein [EGZ11] is not complete. We provide here an alternative proof, valid in the ample locus of any big cohomology class, and discuss the resulting modifications.
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 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 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 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.
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.
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.
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 the image under any dominant meromorphic map of the Monge-Amp{è}re measure of a H{ö}lder continuous quasi-psh function still possesses a H{ö}lder potential. We also discuss the case of lower regularity.
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.
We show that the Monge-Ampère density of the extremal function $V_P$ for a non-convex Pac-Man set $P\subset {\bf R}^2$ tends to a finite limit as we approach the vertex $p$ of $P$ linearly but with a value that may vary with the line. On the other hand, along a tangential approach to $p$ the Monge-Ampère density becomes unbounded. This partially mimics the behavior of the Monge-Ampère density of the union of two quarter disks set $S$ of Sigurdsson and Snaebjarnarson. We also recover their formula for $V_S$ by elementary methods.
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…
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.
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…
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.
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.
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.
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.
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 $\mathbb{C}^n$.
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.
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 $\C^n$.
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.
