arXiv research

New estimate for complex Monge-Ampère equations improves previous results.

Gradient and Laplacian estimates for complex Monge-Ampère equations found.

Paper establishes $L^{\infty}$ estimates for complex Monge-Ampere and Hessian equations.

Solves complex Monge-Ampère equation for measures with pluripolar parts.

Paper proves uniform continuity bounds for complex Monge-Ampère solutions.

Note on gradient estimates for complex Monge-Ampere equation.

We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.

We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…

New proof for stability estimates in complex equations without pluripotential theory.

Sharp $L^\infty$ estimates proved for complex Monge-Ampère equations.

Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.

Solves complex Monge-Ampère equations on Kähler manifolds.

Stability results for complex Monge-Ampère equations in various classes.

Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.

Proves projectivity and ampleness of a Kähler manifold using complex Monge-Ampère equation.

Solves complex Monge-Ampère equations with Hölder continuous solutions in Kähler manifolds.

Study complex Monge-Ampère equations on compact Kähler manifolds.

This paper proves Hölder continuity for complex Monge-Ampère equations on Kähler varieties.

Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.

Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.

Inspired by constructions in complex geometry we introduce a thermodynamic framework for Monge-Ampère equations on real tori. We show convergence in law of the associated point processes and explain connections to complex Monge-Ampère equations and optimal transport.

Proves Hölder continuity of complex Monge-Ampère solutions.

In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equatio…

New method solves complex Monge-Ampère equations on hermitian manifolds.

Estimates complex Hessian integral for complex Monge-Ampère equations.

We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein met…

In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution $u$ of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then $u$ is a quadratic polynomial.

Survey on complex Monge-Ampère equations in Hermitian contexts.

Unified estimate for complex Monge-Ampère equations on Kähler manifolds.

We show a very general existence theorem to the complex Monge-Ampère type equation on hyperconvex domains.

Study on positivity properties of vector bundle Monge-Ampère equation.

Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

We show a general existence theorem to the complex Monge-Ampère type equation on compact Kähler manifolds.

We study the parabolic complex Monge-Ampère type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a way to carry out method of continuity for elliptic Monge-Ampére type equations.

We show that, up to scaling, the complex Monge-Ampere equation on compact Hermitian manifolds always admits a smooth solution.

We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.

We introduce generalized Monge-Ampère capacities and use these to study complex Monge-Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor.

Researchers find explicit solutions to complex Monge-Ampère equation.

Study bounds on Monge-Ampère volumes for degenerate complex equations.

We consider the complex Monge-Ampère equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.

We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.

Auxiliary equations improve bounds in symplectic geometry.

Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

Uniform estimates for complex Monge-Ampère equations on hermitian varieties.

We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced…

We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

We consider the complex Monge-Ampére equation on complete Kähler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some \emph{weighted} $W^{1,p_0}$ space for $p_0 > 2n$, the Monge-Ampére equation has a classical $W^…