$C^{1,1}$ regularity of degenerate complex Monge-Amp\`{e}re equations and some applications

TL;DR

This paper establishes a $C^{1,1}$ regularity estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds, enabling existence results and applications to geometric problems like Sasakian metric geodesics.

Contribution

It provides the first $C^{1,1}$ regularity estimate for degenerate complex Monge-Ampère equations on almost Hermitian manifolds, with broad applications.

Findings

Proved $C^{1,1}$ estimates for solutions on compact almost Hermitian manifolds.

Established existence of $C^{1,1}$ solutions to degenerate and singular Monge-Ampère equations.

Applied results to the regularity of geodesics in Sasakian geometry.

Abstract

In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp\`{e}re equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Amp\`{e}re equations, the corresponding Dirichlet problems and the singular Monge-Amp\`{e}re equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge-Amp\`{e}re type equations. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of Sasakian metrics.