Regularity properties of the degenerate Monge--Ampère equations compact Kähler manifolds

Regularity estimates of solutions to complex Monge-Ampère equations on Hermitian manifolds

In this paper, we obtain the Bedford-Taylor interior $C^{2}$ estimate and local Calabi $C^{3}$ estimate for the solutions to complex Monge-Ampère equations on Hermitian manifolds.

A remark on the quaternionic Monge-Ampère equation on foliated manifolds

We study the quaternionic Monge-Ampère equation on HKT manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Ampère equation has always a unique solution for every basic datum. This approach includes the study of the equation on SU(3).

The complex Monge-Ampére equation on the complement of a divisor

We consider the complex Monge-Ampére equation on complete Kähler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some \emph{weighted} $W^{1,p_0}$ space for $p_0 > 2n$, the Monge-Ampére equation has a classical $W^{3,p_0}$ solution.

Second-order generalized Monge--Ampère equations on a plane and its geometric singular solutions

In the present paper, we study some generalized Monge--Ampère equations in terms of special exterior differential systems on a jet space. Moreover, we construct geometric singular solutions of the generalized Monge--Ampère equations by using the method of Cauchy characteristics. Furthermore, we give criteria for geometric singular solutions to be right-left equivalent to the cuspidal edge, swallowtail and butterfly.

关于Kähler-Ricci流的Ricci曲率

On the Ricci curvature of Kahler-Ricci Flow

本文研究Kähler-Ricci流中Kodaira维数为一的流形的Ricci曲率收敛性，揭示其在局部拓扑下的收敛行为。

Integrable geometries and Monge-Ampere equations

In this lecture delivered at the Integrable and Quantum Field Theory at Peyresq sixth meeting, we review the Lychagin's Monge-Ampere operators theory and exhibit the link it establishes between the classical problem of local equivalence for non linear partial differential equations and the problem of integrability of some geometrical structures.

A mean value formula and a Liouville theorem for the complex Monge-Ampère equation

In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex Monge-Ampère equation on product manifolds.

Monge-Ampere equations and generalized complex geometry. The two-dimensional case

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.

大上同调类中蒙日-安培势的唯一性与稳定性

Uniqueness and Stability of Monge--Ampère Potentials in Big Cohomology Classes

本文针对大上同调类中退化复蒙日-安培方程的解建立了稳定性与一致估计，进而证明了参数充分小时复蒙日-安培平均场方程解的唯一性。

Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds

We prove $C^\infty$ convergence for suitably normalized solutions of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti and Weinkove.

Some multi-valued solutions to Monge-Ampere equations

We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.

Energy Functionals for the Parabolic Monge-Ampere Equation

We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.

Some remarks on contact manifolds, Monge-Ampere equations and solution singularities

We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on some aspects of contact geometry.

The Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds

In this paper, we solve the Dirichlet problem for Monge-Ampère type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.

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.

Parabolic Complex Monge-Ampère Type Equations on Closed Hermitian Manifolds

We study the parabolic complex Monge-Ampère type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a way to carry out method of continuity for elliptic Monge-Ampére type equations.

A new gradient estimate for the complex Monge-Ampère equation

A gradient estimate for complex Monge-Ampère equations which improves in some respects on known estimates is proved using the ABP maximum principle.

A Liouville Theorem for the Complex Monge-Ampère Equation

In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution $u$ of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then $u$ is a quadratic polynomial.

Application of equivalence method to Monge-Ampère equations: Elliptic case

The application of equivalence method to classify Monge-Ampère system leads to three orbits, parabolic case, hyperbolic case and elliptic case wich correspond to three types of Monge-Ampère systems. In this paper we will study the elliptic case and give a presentation of the group as a complex group.

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.

Calibrations associated to Monge-Ampere equations

We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.

The parabolic Monge-Ampère equation on compact almost Hermitian manifolds

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of the Monge-Ampère equation. This gives a parabolic proof of existence of solutions to the Monge-Ampère equation on almost Hermitian manifolds.

Generalized complex Monge-Ampère type equations on closed Hermitian manifolds

We study generalized complex Monge-Ampère type equations on closed Hermitian manifolds. We derive {\em a priori} estimates and then prove the existence of admissible solutions. Moreover, the gradient estimate is improved.

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.

Parabolic Flow for Generalized complex Monge-Ampère type equations

We study the parabolic flow for generalized complex Monge-Ampère type equations on closed Hermitian manifolds. We derive {\em a priori} $C^\infty$ estimates for normalized solutions, and then prove the $C^\infty$ convergence.

Geometric estimates for complex Monge-Ampere equations

We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein metrics on Kahler manifolds of nonnegative Kodaira dimensions.

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.

Comparison of Monge-Ampère capacities

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.

Symmetry Results for classical solutions of Monge-Ampere systems on a bounded planar domain

In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.

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.

Counterexamples to the Local Solvability of Monge-Ampere Equations in the Plane

In this paper, we present smooth examples of degenerate hyperbolic and mixed type Monge-Ampere equations in the plane, which do not admit a local C^3 solution.

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.

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampère equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampère operators. We then use the Perron method to solve Monge-Ampère equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoë, Caffarelli-Viaclovsky and Hultgren-Önnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow…

抛物型分裂型蒙日-安培方程在分裂切丛表面上的研究

The parabolic split-type Monge-Ampere on split tangent bundle surfaces

本文研究了抛物型分裂型蒙日-安培方程的长期存在性及收敛性，探讨了其在凯勒背景下的应用及在曲率条件下对扭曲蒙日-安培方程的简化。

Local noncollapsing for complex Monge-Ampère equations

We prove a local volume noncollapsing estimate for Kähler metrics induced from a family of complex Monge-Ampère equations, assuming a local Ricci curvature lower bound. This local volume estimate can be applied to establish various diameter and gradient estimate.

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.

Stability estimates for the complex Monge-Ampère and Hessian equations

A new proof for stability estimates for the complex Monge-Ampère and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Ampère equation, and under degenerations to a big class in the case of Hessian equations.

Regularity of weak solutions of a complex Monge-Ampère equation

We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.

Expansion formula for complex Monge-Ampère equation along cone singularities

In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Ampère equation which arise naturally in the study of the conical Kähler-Einstein metric.

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.

退化复蒙日-安培方程 第一部分

Degenerate Complex Monge Ampere Equation Part I

该研究解决了自丘成桐奠基性工作以来悬而未决的问题，刻画了Kähler流形上退化复Monge-Ampère方程解的奇异性态。

The Monge-Ampère equation for non-integrable almost complex structures

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

Monge-Ampère type equations on almost Hermitian manifolds

In this paper, we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.

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.

Symmetry Results for classical solutions of Monge-Ampere system in the plane

In this paper, by the method of moving planes, we prove the symmetry result which says that classical solutions of Monge-Ampere system in the whole plane are symmetric about some point. Our system under consideration comes from the differential geometry problem.

Nakano正曲率张量的全纯向量丛的Monge-Ampère类型方程

Monge-Ampère type equation for the Nakano positive curvature tensor of holomorphic vector bundles

本文研究了全纯向量丛的Nakano正曲率张量的Monge-Ampère类型方程，并在共形类中解决了该方程。

On Pointwise Gradient Estimates for the Complex Monge-Ampere Equation

In this note, a gradient estimate for the complex Monge-Ampere equation is established. It differs from previous estimates of Yau, Hanani, Blocki, P. Guan, B. Guan - Q. Li in that it is pointwise, and depends only on the infimum of the solution instead of its $C^0$ norm.

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…

通过积分方法估计复奇异 Monge-Ampère 方程

Estimates for complex singular Monge-Ampère equations via integral method

针对奇异复 Monge-Ampère 方程，本文采用积分方法推导其解的梯度与 Laplacian 估计。

A gradient estimate for the Hermitian Monge-Ampère equation

We improve our previous gradient estimate for the Monge-Ampère equation on a compact Hermitian manifold and give a estimates for the non-mixed second order derivatives. These estimates are required to apply either the Evans-Krylov estimates for second order derivatives or the third derivatives estimates for equations with a gradient term.

通过倍增方法研究Monge-Ampère方程的正则性

Regularity for the Monge-Ampère equation by doubling

本文利用极大Lagrangian子流形上Hessian的倍增不等式，为Monge-Ampère方程严格凸解的内部正则性提供了一个新证明。