arXiv research

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…

Solves complex Monge-Ampère equation for measures with pluripolar parts.

The study identifies Hermitian metrics preserving the total Monge-Ampere volume.

Proves existence of eigenvalue and eigenfunction for complex Monge-Ampère operator.

Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.

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 la…

Study complex Monge-Ampère operator on weighted pluricomplex energy classes.

Study complex Monge-Ampère equations on compact Kähler manifolds.

The paper studies quaternionic Monge-Ampère equations in weighted energy classes.

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…

We introduce a wide subclass ${\cal F}(X,ω)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well-defined and the convergence theorem is valid. We also prove that ${\cal F}(X,ω)$ is a convex cone and includes all quasi-plurisubharmonic functions which are …

We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.

This paper develops Lagrangian potential theory and a Monge-Ampère operator.

Study solves complex equation on specific types of manifolds.

The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.

The paper finds transformation formulas for quaternionic complex structures.

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibilit…

Let $(X,ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,ω)$ of the complex Monge-Am…

We define a non-degenerated Monge-Ampere structure on a 6-manifold associated with a Monge-Ampere equation as a couple (Ω,ω), such that Ωis a symplectic form and ωis a 3-differential form which satisfies ω\wedgeΩ=0 and which is non-degenerated in the sense of Hitchin. We associate with such a couple an almost (pseudo) …

Study estimates eigenvalues for concave Hessian operators on convex domains.

We define a nondegenerate Monge-Ampère structure on a 6-dimensional manifold as a pair $(Ω,ω)$, such that $Ω$ is a symplectic form and $ω$ is a 3-differential form which satisfies $ω\wedgeΩ=0$ and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau stru…

General existence theorem for complex Monge-Ampère on hyperconvex domains.

Existence theorem for complex Monge-Ampère on compact Kähler manifolds.

New estimate for complex Monge-Ampère equations improves previous results.

Uniform estimates for complex Monge-Ampere equations help understand singular solutions and collapsing metrics.

Let $X$ be a compact Kähler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are …

Use envelope method to prove existence of complex Monge-Ampère equations on compact Kähler manifolds.

Gradient and Laplacian estimates for complex Monge-Ampère equations found.

Paper proves $C^{1,1}$ regularity for complex Monge-Ampère equations.

Uniform estimates for complex equations on compact manifolds found.

Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.

We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.

Paper proves uniform continuity bounds for complex Monge-Ampère solutions.

New geometric perspective for optimal learning on hexagonal structures.

We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.

Note on gradient estimates for complex Monge-Ampere equation.

Paper establishes $L^{\infty}$ estimates for complex Monge-Ampere and Hessian equations.

The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.

Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.

Proves a smoothness estimate for complex Monge-Ampère solutions.

We prove Aubin's "Hypothese fondamentale" concerning the existence of Moser-Trudinger type inequalities on any integral compact Kähler manifold X. In the case of the anti-canonical class on a Fano manifold the constants in the inequalities are shown to only depend on the dimension of X (but there are counterexamples to…

Solves complex Monge-Ampére equation with weakly regular right-hand side.

Solves complex Monge-Ampère equation with Hölder continuous boundary data.

Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.

The paper proves a formula for complex Monge-Ampère equations on manifolds.

Stability results for complex Monge-Ampère equations in various classes.

Proves Hölder continuity of complex Monge-Ampère solutions.

Sharp $C^0$ bound for quaternionic Monge-Ampere on hyperhermitian manifolds.