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A new gradient estimate for the complex Monge–Ampère equation
A gradient estimate for complex Monge–Ampère equations which improves in some respects on known estimates is proved using the ABP maximum principle.
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The Eigenvalue Problem for the Complex Monge–Ampère Operator
We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge–Ampère operator on a...
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Quaternionic Monge–Ampère Measure on Pluripolar Set
In this paper, we prove that in a hyperconvex domain Ω in ℍ n , if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure,...
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Gradient and Hessian Estimates for the Hermitian Monge–Ampère Equation
We substantially improve our previous gradient estimate for the Monge–Ampère equation on a compact Hermitian manifold. The improvements concern the...
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Stability estimates for the complex Monge-Ampère and Hessian equations
A new proof for stability estimates for the complex Monge-Ampère and Hessian equations is given, which does not require pluripotential theory. A...
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Uniform Estimates for Concave Homogeneous Complex Degenerate Elliptic Equations Comparable to the Monge-Ampère Equation
We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings...
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Monge–Ampère operators and valuations
Two classes of measure-valued valuations on convex functions related to Monge–Ampère operators are investigated and classified. It is shown that the...
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On the Deformations of Symplectic Structure Related to the Monge–Ampère Equation on the Kähler Manifold P2(ℂ)
We analyze the cohomology structure of the fundamental two-form deformation related to a modified Monge–Ampère type on the complex Kähler manifold P 2 (ℂ)...
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Continuity of Monge–Ampère Potentials with Prescribed Singularities
We study the continuity of solutions to complex Monge–Ampère equations with prescribed singularity type. This generalizes the previous results of Di...
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A Subsolution Theorem for the Monge-Ampère Equation over an Almost Hermitian Manifold
Let Ω ⊆ M be a bounded domain with a smooth boundary ∂Ω, where ( M, J, g ) is a compact, almost Hermitian manifold. The main result of this paper is to...
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On Quasilinearization of Elliptic Monge–Ampère Systems
AbstractThe paper deals with finding of the necessary and sufficient conditions for the local quasilinearizability of elliptic Monge–Ampère systems.
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Curvature of the base manifold of a Monge–Ampère fibration and its existence
In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge–Ampère equation, which is called a Monge–Ampère...
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Monge—Ampère Type Equations on Almost Hermitian Manifolds
In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C ∞ a priori estimates under the existence...
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The Dirichlet problem for the Monge–Ampère equation on Hermitian manifolds with boundary
We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge–Ampère equation on a general Hermitian manifold with...
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Weak Solutions to Monge–Ampère Type Equations on Compact Hermitian Manifold with Boundary
We prove the bounded subsolution theorem for the complex Monge–Ampère type equation, with the right-hand side being a positive Radon measure, on a...
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On Large Deviation Principles and the Monge–Ampère Equation (Following Berman, Hultgren)
This is mostly an exposition, aimed to be accessible to geometers, analysts, and probabilists, of a fundamental recent theorem of R. Berman with...
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A Class of Singular Coupled Systems of Superlinear Monge-Ampère Equations
In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by...
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On Lelong numbers of generalized Monge–Ampère products
We consider generalized (mixed) Monge–Ampère products of quasiplurisubharmonic functions (with and without analytic singularities) as they were...
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Geodesic Distance and Monge—Ampère Measures on Contact Sets
We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef...