Related papers: Generalized complex Monge-Amp\`ere type equations …

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…

Uniform bounds are obtained using the auxiliary Monge-Amp\`ere equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G.…

Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a K\"ahler form on $X$. We consider the complex Monge-Amp\`ere equation $(dd^c u+\omega)^n=\mu$, where $\mu$ is a given positive measure on $X$ of suitable mass and $u$ is…

We prove a priori estimates for a generalised Monge-Amp\`ere PDE with "non-constant coefficients" thus improving a result of Sun in the K\"ahler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to…

We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…

In this paper, we establish local potential estimates and H\"older estimates for solutions of linearized Monge-Amp\`ere equations with the right-hand side being a signed measure, under suitable assumptions on the data. In particular, the…

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…

We study asymptotic behaviors of solutions to the Monge-Amp\`ere equation in cones and use the expansion as a tool to study the regularity of solutions in polygonal domains.

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues…

We study the regularity of solutions to complex Monge-Amp\`ere equations $(dd^c u)^n=f dV$, on bounded strongly pseudoconvex domains $ \Omega \subset \C^n$. We show, under a mild technical assumption, that the unique solution $u$ to such an…

We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function…

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the…

We generalize and strenghten Ko{\l}odziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Amp\`ere equation.

Let $X$ be a compact K\"ahler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_{\chi}(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Amp\`ere…

We describe the behavior of the singularities of solutions to degenerate complex Monge-Amp`ere equations on K\"ahler manifolds. This was not resolved since the fundamental paper of S-T Yau \cite{y} on this subject.

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…

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-Amp\`ere…

We study a Monge-Amp\`{e}re equation with power term for some $p\in{\mathbb{R}}$. A solution $u$ is called to be Euclidean complete if it is an entire solution defined over the whole ${\mathbb{R}}^n$ or its graph is a large hypersurface…

We consider a 3rd-order generalized Monge-Ampere equation u_yyy - u_xxy^2 + u_xxx u_xyy = 0 (which is closely related to the associativity equation in the 2-d topological field theory) and describe all integrable structures related to it…

Existence and boundary regularity away from the corners are established for two-dimensional Monge-Amp\`{e}re equations on convex polytopes with Guillemin boundary conditions. An important step is to derive an expansion in terms of functions…