Related papers: Weak solutions to degenerate complex Monge-Amp\'er…
Following the recent development by Guo-Phong-Tong and Chen-Cheng, we derived the $L^{\infty}$ estimate for K\"ahler-Ricci flows under a weaker assumption. The technique also extends to more general cases coming from different geometric…
We compare various notions of weak subsolutions to degenerate complex Monge-Amp{\`e}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions.…
In an earlier work joint with X. X. Chen and G. Tian, we introduced the weak K\"ahler-Ricci flow for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is…
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.
In this article we study the K\"ahler Ricci flow, the corresponding parabolic Monge Amp\`{e}re equation and complete non-compact K\"ahler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$ is…
We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Amp\`ere equations. This type of equations is precisely what is needed in order to construct K\"ahler-Einstein metrics over…
We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Amp\`ere flow on a compact K\"ahler manifold, with the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere…
Given a compact K\"ahler manifold, we survey the study of complex Monge-Amp\`ere type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of…
We establish a stability result for elliptic and parabolic complex Monge-Amp{\`e}re equations on compact K{\"a}hler manifolds, which applies in particular to the K{\"a}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of…
We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})-Au+f(z,t)$ 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 where the boundary condition is…
We study families of complex Monge-Amp\`ere equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\infty}$-estimates for the normalized solutions, generalizing the…
In this note, we study a degenerate twisted J-flow on compact K\"ahler manifolds. We show that it exists for all time, it is unique and converges to a weak solution of a degenerate twisted J-equation. In particular, this confirms an…
The Dirichlet problem for complex Monge-Amp\'ere equations with continuous data is considered. In particular, a notion of viscosity solutions is introduced; a comparison principle and a solvability theorem are proved; the equivalence…
In this paper, we prove that any solution of K\"ahler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no K\"ahler-Einstein metrics. As an application, we found two Fano…
Consider the Kahler-Ricci flow with finite time singularities over any closed Kahler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by…
We study various capacities on compact K\"{a}hler manifolds which generalize the Bedford-Taylor Monge-Amp\`ere capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Amp\`ere…
The aim of this paper is to further develop the theory of the degenerate complex Hessian equations on compact Hermitian manifolds. Building upon the generalization of the Bedford-Taylor pluripotential theory to complex Hessian equations by…
We prove uniform sup-norm estimates for the Monge-Ampere equation with respect to a family of Kahler metrics which degenerate towards a pull-back of a metric from a lower dimensional manifold. This is then used to show the existence of…
The regularity theory of the degenerate complex Monge-Amp\`{e}re equation is studied. The equation is considered on a closed compact K\"{a}hler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a…
The Monge-Amp\`ere type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Amp\`ere type equations, which can be degenerate or singular near…