On the modulus of continuity of solutions to complex Monge-Amp\`ere equations

Bin Guo; Chuwen Wang; Duong H. Phong; Freid Tong

On the modulus of continuity of solutions to complex Monge-Amp\`ere equations

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arxiv 2112.02354 v1 pith:REX5JMUC submitted 2021-12-04 math.DG math.APmath.CV

On the modulus of continuity of solutions to complex Monge-Amp\`ere equations

Bin Guo , Duong H. Phong , Freid Tong , Chuwen Wang This is my paper

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keywords equationsmonge-ampahlerapproachcomplexcontinuitymodulussolutions

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In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Amp\`ere equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates for fully non-linear equations in K\"ahler geometry. As an application, we derive a uniform diameter bound for K\"ahler metrics satisfying certain Monge-Amp\`ere equations.

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Complex Monge-Amp\`ere equation in Orlicz space and Diameter Bound
math.DG 2026-01 conditional novelty 6.0

Under general Orlicz-space integrability of the Monge-Ampère measure, the paper proves L∞ and stability estimates for the potential and derives uniform diameter and volume lower bounds for the associated Kähler metrics.