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arXiv · 2610.39259

Uniform estimates for complex Monge-Ampère equations II: Hermitian manifolds

Quang-Tuan Dang·Lei Zhang·Bin Zhou
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Abstract

We establish uniform estimates for degenerate complex Monge--Ampère equations on compact Hermitian manifolds under weak Orlicz integrability assumptions on the densities. Our estimates apply uniformly to Hermitian metrics and to semipositive uniformly non-collapsing reference forms. For arbitrary big reference forms, we obtain uniform relative $L^\infty$-estimates with respect to the extremal function with minimal singularities, and establish continuity of the resulting solutions on the regular locus. As an application, we prove the existence of solutions with minimal singularities to degenerate complex Monge--Ampère equations associated with big forms. The proofs combine local Orlicz estimates with pluripotential-theoretic envelope techniques and uniform control of the Monge--Ampère masses, overcoming the lack of invariance of the total Monge--Ampère mass in the Hermitian setting.

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Quang-Tuan Dang, Lei Zhang, Bin Zhou. 2026-10-29. Uniform estimates for complex Monge-Ampère equations II: Hermitian manifolds. https://arxiv.org/abs/2610.39259

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