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arXiv · 2410.04395

Complex Alexandrov-Bakelman-Pucci estimate and its applications

Junbang Liu
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Abstract

We prove an Alexandrov-Bakelman-Pucci type estimate, which involves the integral of the determinant of the complex Hessian over a certain subset. It improves the classical ABP estimate adapted (by inequality $2^{2n}|\det(u_{i\bar{j}})|^2\geq |\det(\nabla^2u)|$) to complex setting. We give an application of it to derive sharp gradient estimates for complex Monge-Amp\`ere equations. The approach is based on the De Giorgi iteration method developed by Guo-Phong-Tong for equations of complex Monge-Amp\`ere type.

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Junbang Liu. 2024-10-06. Complex Alexandrov-Bakelman-Pucci estimate and its applications. https://arxiv.org/abs/2410.04395

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