arXiv research

New geometries derived from symplectic Monge-Ampère structures.

The paper generalizes Monge-Ampère equations and their solutions in differential geometry.

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

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

Unique solution found for quaternionic Monge-Ampère equation on specific HKT manifolds.

Study proves long-term solutions to a specific equation on hyperKähler manifolds.

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

Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.

Develops a new parabolic equation for surfaces, proving long-time existence and convergence.

Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.

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 Monge-Ampere operator in the class of w-plurisubharmonic functions with vanishing…

Study compares Monge-Ampère capacities on Kähler manifolds.

The paper studies quaternionic Monge-Ampère equations in weighted energy classes.

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

Study geometric singular solutions of generalized Monge-Ampère equations.

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

Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.

We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.

New proof for convex solutions of Monge-Ampère equation.

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

The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.

Note on gradient estimates for complex Monge-Ampere equation.

New proof of $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes.

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

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.

Study complex Monge-Ampère equations on compact Kähler manifolds.

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

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

We investigate a class of multi-dimensional two-component systems of Monge-Ampère type that can be viewed as generalisations of heavenly-type equations appearing in self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of skew-symmetric matrix pencils, a classification of normal forms of such systems is o…

Enhances gradient estimates for Hermitian Monge-Ampère equations.

Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.

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…

Solves a specific Dirichlet problem on Hermitian manifolds.

In this note, we describe the Hermitian metrics that leave the total Monge-Ampere volume invariant. In particular, we give several characterizations of the Hermitian metrics which satisfy the comparison principle for the complex Monge-Ampere operator

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

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of t…

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.

Solves a general class of free boundary Monge-Ampère equations.

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

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

Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.

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.

We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on some aspects of contact geometry.

Proves Hölder continuity of complex Monge-Ampère solutions.

Study of pseudo-Riemannian metrics related to Monge-Ampère structures.

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

We study degenerate complex Monge-Ampère equations on a compact Kähler manifold $(X,ω)$. We show that the complex Monge-Ampère operator $(ω+ dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,ω)$ of $ω$-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class ${\mathcal E}(X,ω)$ is the la…

Study on new Monge-Ampère functionals and their variational problems.