相关论文: Monge-Ampere equations and generalized complex geo…

In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Amp\`ere equation which arise naturally in the study of the conical K\"ahler-Einstein metric.

We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity…

It is shown that the Monge equation is equivalent to the ordinary differential equation $\ddot X=0$ of free motion. Equations of Monge type (with their general solutions) are connected with each ordinary differential equation of second…

We consider the complex Monge-Amp\`ere equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.

In this paper we consider symplectic and contact Lie algebras. We define contactization and symplectization procedures and describe its main properties. We also give classification of such algebras in dimensions 3 and 4. The classification…

The partial Legendre transform of a non-linear elliptic differential equation is shown to be another non-linear elliptic differential equation. In particular, the partial Legendre transform of the Monge-Amp\`ere equation is another equation…

This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory,…

In a recent paper, Darvas-Rubinstein proved a convergence result for the Kahler-Ricci iteration, which is a sequence of recursively defined complex Monge-Ampere equations. We introduce the Monge-Ampere iteration to be an analogous, but more…

We consider a generalised complex Monge-Amp\`ere equation on a compact K\"ahler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related…

We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…

The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the…

Monge-Amp\`ere equations of the form, $u_{xx}u_{yy}-u_{xy}^2=F(u,u_x,u_y)$ arise in many areas of fluid and solid mechanics. Here it is shown that in the special case $F=u_y^4f(u, u_x/u_y)$, where $f$ denotes an arbitrary function, the…

A number of geometric problems, including affine hyperbolic spheres, Hilbert metrics and Minkowski type problems, are reduced to a singular Monge-Amp\`ere equation which can be written locally as a class of Monge-Amp\`ere equations with…

In this paper, we study the Neumann problem of Monge-Amp\`ere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension $n\geq 3$, we show that…

This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Amp\`ere equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e.…

Let u be a function of n independent variables x^1, ..., x^n, and U=(u_{ij}) the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a linear relation among all possible minors of U. Particular examples include the…

In this note, we give a proof of the uniform log-continuity of the solution to complex Monge-Amp\`ere equations on compact Hermitian manifolds, which is a generalization of the result of Guo-Phong-Tong-Wang in the K\"ahler case.

This is mostly an exposition, aimed to be accessible to geometers, analysts, and probabilists, of a fundamental recent theorem of R. Berman with recent developments by J. Hultgren, that asserts that the second boundary value problem for the…

In this paper, we obtain gradient estimates and Laplacian estimates for the solution to the singular complex Monge-Amp\`ere equation by applying the integral method.

In this paper, we explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge--Amp\`ere equation. Among the results we obtain through these connections are: $(i)$~a theorem on the continuous extension…