On relative $L^\infty$ estimate for complex Monge-Amp\`ere equations

Junbang Liu

On relative $L^\infty$ estimate for complex Monge-Amp\`ere equations

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arxiv 2410.04393 v1 pith:O2TIC2HN submitted 2024-10-06 math.DG math.AP

Junbang Liu This is my paper · ORCID

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abstract

We prove a relative $L^\infty$ estimate for a class of complex Monge-Amp\`ere type equations on K\"ahler manifolds. It provides a unified approach to Tundinger type estimate and uniform estimate. It also improves the previous results about modulus of continuity, stability estimates, and $W^{1,1}$-estimates of Green's functions. The argument is based on the PDE method developed by Guo-Phong-Tong and constructing appropriate comparison metrics from entropy bound.

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Complex Monge-Amp\`ere equation in Orlicz space and Diameter Bound
math.DG 2026-01 conditional novelty 6.0 of 10

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.

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