Complex geodesics and complex Monge--Amp\`{e}re equations with boundary singularity II

TL;DR

This paper investigates the relationship between boundary regularity and solutions to complex Monge--Ampère equations with boundary singularities, providing improved results on the regularity of the pluricomplex Poisson kernel in convex domains.

Contribution

It establishes a quantitative link between the regularity of the pluricomplex Poisson kernel and the boundary smoothness of strongly linearly convex domains, advancing previous work.

Findings

Improved regularity estimates for the pluricomplex Poisson kernel.

Quantitative relationship between boundary smoothness and Monge--Ampère solutions.

Enhanced understanding of complex geodesics with boundary conditions.

Abstract

We study the parameter dependence of complex geodesics with prescribed boundary value and direction on bounded strongly linearly convex domains. As an important application we establish a quantitative relationship between the regularity of the pluricomplex Poisson kernel of such a domain, which is a solution to a homogeneous complex Monge--Amp\`{e}re equation with boundary singularity, and the regularity of the boundary of the domain. Our results greatly improve the previous results of Chang--Hu--Lee and Bracci--Patrizio in this direction.