Some multi-valued solutions to Monge-Ampere equations

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.

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.

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.

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-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.

Homogeneous Backlund transformations of hyperbolic Monge-Ampere systems

A Backlund transformation between two hyperbolic Monge-Ampere systems may be described as a certain type of exterior differential system on a 6-dimensional manifold B. The transformation is homogeneous if the group of symmetries of the system acts transitively on B. We give a complete classification of homogeneous Backlund transformations between hyperbolic Monge-Ampere systems.

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.

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.

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.

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。

$C^{1,1}$ regularity of degenerate complex Monge-Ampère equations and some applications

In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge-Ampère type equations. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of …

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.

Permanental Point Processes on Real Tori, Theta Functions and Monge-Ampère Equations

Inspired by constructions in complex geometry we introduce a thermodynamic framework for Monge-Ampère equations on real tori. We show convergence in law of the associated point processes and explain connections to complex Monge-Ampère equations and optimal transport.

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.

超Kähler流形上的抛物型四元数Monge-Ampère型方程

The parabolic quaternionic Monge-Ampère type equation on hyperKähler manifolds

本文证明紧致超Kähler流形上抛物型四元数Monge-Ampère型方程解的长时存在唯一性，并证明归一化后光滑收敛到相应Monge-Ampère方程的唯一解。

Smoothness of radial solutions to Monge-Ampere equations

We characterize when radial weak solutions to Monge-Ampere equations are smooth. This paper extends previous partial results and also covers Generalized Monge-Ampere equations and infinitely vanishing right hand side.

The $L^{\infty}$ estimates for parabolic complex Monge-Ampere and Hessian equations

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.

Gradient estimate for complex Monge-Ampere equation with continuous right hand side

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.

On the modulus of continuity of solutions to complex Monge-Ampère equations

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.

Global $W^{2,p}$ estimates for the Monge-Ampere equation

We obtain global $W^{2,p}$ estimates for the Monge-Ampere equation under natural assumptions on the domain and boundary data.

The Monge-Ampère equation and its link to optimal transportation

We survey the (old and new) regularity theory for the Monge-Ampère equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Ampère type equations arising in that context.

Interior $C^2$ estimate for Monge-Ampère equation in dimension two

We obtain a genuine local $C^2$ estimate for the Monge-Ampère equation in dimension two, by using the partial Legendre transform.

On the Monge-Ampère equation

This text contains the material discussed by the author in the Bourbaki seminar of June 2018, on the recent developments in the theory of the Monge-Ampère equation.

Boundary regularity for solutions to the linearized Monge-Ampère equations

We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues of the boundary Holder gradient estimates of Krylov.

A Monge-Ampere Type Fully Nonlinear Equation on Hermitian Manifolds

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.

刻画复Monge-Ampère算子的值域

Characterizing the range of the complex Monge-Ampère operator

本文在紧Kähler流形上求解具有多极部分的测度的复Monge-Ampère方程，推广了Cegrell的经典结果，并探讨了算子的特殊性质。

On regularity of complex Monge-Ampere equation

We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for Laplacian u, and of Lipschitz condition on right hand side. Then we shall construct a family of Pogorelov-type examples for complex Monge-Ampere equation. These examples give generalized entire solutions (as well as viscosity solutions) of complex Monge-Ampere equation $\det(u_{i\bar j})=1.

Asymptotic expansion at infinity of solutions of Monge-Ampère type equations

We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Ampère type equations that origin from mean curvature equations of Lagrangian graph $(x,Du(x))$ and refine the previous study on zero mean curvature equations and the Monge-Ampère equations.

复Monge-Ampère型方程的边界情形

The boundary case for complex Monge-Ampère type equations

该研究聚焦复Monge-Ampère型方程的边界情形问题，在特定几何假设下开展相关分析，为该类方程的边界相关理论研究提供参考。

Boundary Harnack inequality for the linearized Monge-Ampère equations and applications

In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez's interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upper bound and global $L^p$-integrability for the Green's function of the linearized Monge-Ampère operator.

A Liouville's theorem for some Monge-Ampère type equations

We study a Monge-Ampère type equation that interpolates the classical {σ_2} -Yamabe equation in conformal geometry and the 2-Hessian equation in dimension 4.

Pointwise $C^{2,α}$ estimates at the boundary for the Monge-Ampere equation

We obtain pointwise $C^{2,α}$ estimates at boundary points for solutions to the Monge-Ampere equation under appropriate local conditions on the right hand side and boundary data.

Global $W^{2,p}$ estimates for solutions to the linearized Monge--Ampère equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Ampère equations.

Entire invariant solutions to Monge-Ampere equations

We prove existence and regularity of entire solutions to Monge-Ampere equations invariant under an irreducible action of a compact Lie group.

A priori estimate for the complex Monge-Ampere equation

In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.

Moser-Trudinger inequality for the complex Monge-Ampere equation

In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.

Non linear elliptic theory and the Monge-Ampere equation

The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ampere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations.

An obstacle problem for a class of Monge-Ampère type functionals

In this paper we study an obstacle problem for Monge-Ampère type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation.

Optimisation approach for the Monge-Ampere equation

This paper studies the numerical approximation of solution of the Dirichlet problem for the fully nonlinear Monge-Ampere equation. In this approach, we take the advantage of reformulation the Monge-Ampere problem as an optimization problem, to which we associate a well defined functional whose minimum provides us with the solution to the Monge-Ampere problem after resolving a Poisson problem by the finite element Galerkin method. We present some numerical examples, for which a good approximation is obtained in 68 iterations.

锥体与多边形区域中的Monge-Ampère方程

The Monge-Ampère equation in cones and polygonal domains

本文研究锥体中Monge-Ampère方程解的渐近行为，并利用该展开工具分析多边形区域内解的正则性。

On boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations

In this paper, we establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,γ}$ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampère measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampère equations with bounded right hand side and $C^{1, 1}$ boundary data.

W^{2,1} estimate for singular solutions to the Monge-Ampere equation

We prove an interior $W^{2,1}$ estimate for singular solutions to the Monge-Ampere equation, and construct an example to show our results are optimal.