Related papers: The complex Monge-Ampere equation on compact Kaehl…

We study the regularizing properties of complex Monge-Amp\`ere flows on a K\"ahler manifold $(X,\omega)$ when the initial data are $\omega$-psh functions with zero Lelong number at all points. We prove that the general Monge-Amp\`ere flow…

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.

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,\alpha} estimates away from a divisor are obtained, by combining…

Let $X$ be a compact K\"ahler manifold whose anticanonical cohomology class is semipositive. Let $L$ be a big and semi-ample line bundle on $X$ and $\alpha$ be the Chern class of $L$. We give a sufficient condition ensuring that the…

We prove the existence of canonical tubular neighbourhoods around complex submanifolds of K\"ahler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Amp\`ere…

Let $(X,\omega)$ be a compact Hermitian manifold and let $\{\beta\}\in H^{1,1}(X,\mathbb R)$ be a real $(1,1)$-class with a smooth representative $\beta$, such that $\int_X\beta^n>0$. Assume that there is a bounded $\beta$-plurisubharmonic…

In this paper, we consider the Monge-Amp\`{e}re type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence…

In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact K\"ahler manifold. We prove that a precise bound on the complex Monge-Amp\`ere…

Uniform $L^\infty$ and H\"older estimates were proved by the Kolodziej for complex Monge-Amp\`ere equations on compact K\"ahler manifolds with $L^p$ volume measure with $p>1$. On the other hand, establishing H\"older estimates on singular…

We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hat{\omega}$-plurisubharmonic solution of an elliptic complex Monge-Amp\`{e}re equation is smooth under an…

In this paper, we establish a local regularity result for $W^{2,p}_{\mathrm{loc}}$ solutions to complex degenerate nonlinear elliptic equations $F(D^2_{\mathbb{C}} u)=f$ when they are comparable to the Monge-Amp\`ere equation. Notably, we…

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 generalize an inequality for mixed Monge-Amp\`ere measures. We also give an example that shows that our assumptions are sharp. The corresponding result in the setting of compact K\"ahler manifold is also discussed.

We study degenerate complex Monge-Amp\`ere equations on a compact K\"ahler manifold $(X,\omega)$. We show that the complex Monge-Amp\`ere operator $(\omega + dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,\omega)$ of…

For the Monge-Amp\`ere equation with a right-hand side bounded away from 0 and infinity, we show that the solution, subject to the natural boundary condition arising in optimal transport, is in $W^{2,1+\varepsilon}$ up to the boundary.

We consider the Dirichlet problem for the complex Monge-Amp\`ere equation in a bounded strongly hyperconvex Lipschitz domain in $\C^n$. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is…

In this paper, we shall study the boundary case for complex Monge-Amp\`ere type equations under certain geometric assumptions.

We discuss pluripotential aspects of the Monge-Amp\`ere equations on compact Hermitian manifolds and prove $L^{\infty}$ estimates for any metric, as well as the existence of weak solutions under an extra assumption.

We study the Dirichlet problem for complex Monge-Ampere equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the flat case. We…

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Amp\`ere equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$…