Related papers: Stability of solutions to complex Monge-Amp\`ere e…
Let $(X,\omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Amp\`ere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are H\"older continuous with…
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…
In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Amp\`ere equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.
We characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Amp\`ere equation has a solution of finite pluricomplex energy. Our results are also valid in the big cohomology class setting.
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.
A new proof for stability estimates for the complex Monge-Amp\`ere 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…
We first establish the weak stability results for solutions of complex Monge-Amp\`ere equations in relative full mass classes, extending the results known to hold in the full mass class. Building on weak stability, we then prove the…
We prove stability of solutions of the complex Monge-Amp\`ere equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be H\"older…
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.
We generalize several known stability estimates for complex Monge-Amp\`ere equations to the setting of low (or high) energy potentials. We apply our estimates to obtain, among other things, a quantitative domination principle, and metric…
On a compact K\"ahler manifold $(X,\omega)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Amp\`ere equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the…
In this work, we study Monge-Ampere equations over closed K\"ahler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.
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…
Extending DiNezza-Lu's approach to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge-Amp{\`e}re equations on compact K{\"a}hler manifolds are continuous on a Zariski open set. This allows us to show…
We show a general existence theorem to the complex Monge-Amp\`ere type equation on compact K\"ahler manifolds.
In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Amp\`ere equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates…
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 generalize previous diameter estimates and local non-vanishing of volumes for Kaehler metrics to the case of big cohomology classes. In our proof, among other things, we will prove a uniform diameter estimate for a family of smooth…
In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Amp\`ere equation with weak initial data. As…
A general solution to the Complex Monge-Amp\`ere equation in a space of arbitrary dimensions is constructed.