Related papers: Asymptotics and convergence for the complex Monge-…
We establish a parabolic version of Tian's $C^{2,\alpha}$-estimate for conical complex Monge-Ampere equations, which includes conical K\"ahler-Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical…
We study the parabolic flow for generalized complex Monge-Amp\`ere type equations on closed Hermitian manifolds. We derive {\em a priori} $C^\infty$ estimates for normalized solutions, and then prove the $C^\infty$ convergence.
This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on K\"ahler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex…
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.
We solve the Dirichlet Problem of Monge-Amp\`ere equation near an isolate Klt singularity, which generalizes the result of Eyssidieux-Guedj-Zeriahi \cite{EGZ}, where the Monge-Amp\`ere equation is solved on singular varieties without…
We prove several quantitative stability estimates for solutions of complex Monge-Ampere equations when both the cohomology class and the prescribed singularity vary. In a broad sense, our results fit well into the study of degeneration of…
We give an explicit lower bound, in terms of the distance from the boundary, for the Kobayashi metric of a certain class of bounded pseudoconvex domains in $\mathbb{C}^n$ with $\mathcal{C}^2$-smooth boundary using the regularity theory for…
In this paper, the author studies quaternionic Monge-Amp\`ere equations and obtain the existence of the solutions to the Dirichlet problem for such equations in strictly pesudoconvex domains in quaternionic space. The stability and…
Precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite- and infinite-dimensional settings. Our results are based on the existence of dominating points for these sets, a related…
We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case of the boundary conditions are…
We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter…
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 deal with Monge-Amp\`ere type equations modeled upon general anisotropic norms $H$ in $\mathbb R^n$. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject to both a homogeneous…
Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The…
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…
We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge-Amp\`ere Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate…
Following the authors' recent work \cite{Zhang-Zhou2025}, we further explore the convexity properties of solutions to the Dirichlet problem for the complex Monge-Amp\`ere operator. In this paper, we establish the $\log$-concavity of…
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,…
We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Amp\`ere equation, the quaternionic Hessian equation and the Monge-Amp\`ere equation for quaternionic $(n-1)$-plurisubharmonic…
We introduce a non local analog to the Monge-Ampere operator and show some of its properties. We prove that a global problem involving this operator has C 1,1 solutions in the full space.