arXiv research

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

Study geometric singular solutions of generalized Monge-Ampère equations.

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

New proof for convex solutions of Monge-Ampère equation.

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…

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

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

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

Researchers find explicit solutions to complex Monge-Ampère equation.

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

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

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

We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.

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.

We study the problem of the existence and the holomorphicity of the Monge-Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a …

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

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

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…

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

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.

Study shows asymptotic behavior of metric near singular points of a Monge-Ampère equation.

We prove the long time existence and uniqueness of solution to a parabolic Monge-Ampère type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge-Ampè…

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

We study the regularizing properties of complex Monge-Ampère flows on a Kähler manifold $(X,ω)$ when the initial data are $ω$-psh functions with zero Lelong number at all points. We prove that the general Monge-Ampère flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solutio…

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.

Proves Hölder continuity of complex Monge-Ampère solutions.

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

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

Proves existence and uniqueness of weak solutions for specific equations.

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.

Solves long-time solutions for a specific equation on hyperkähler manifolds.

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…

The main result asserts the existence of continuous solutions of the complex Monge-Ampère equation with the right hand side in $L^p, p>1$, on compact Hermitian manifolds.

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

We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known to hold in the context of Kähler classes.

In this paper, we obtain the Bedford-Taylor interior $C^{2}$ estimate and local Calabi $C^{3}$ estimate for the solutions to complex Monge-Ampère equations on Hermitian manifolds.

Proves smooth solutions for generalised Monge-Ampère equations on projective manifolds.

Solves complex Monge-Ampère equations on Kähler manifolds.

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.

In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.

New framework for optimal transport with jumps over intermediate spaces.

This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration …

Study on continuity of solutions for complex Monge-Ampère equations with movable singularities.

Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.

Study solves complex equation on specific types of manifolds.

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

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