Isoperimetric inequality for nearly spherical domains in the Bergman ball

TL;DR

This paper establishes a quantitative isoperimetric inequality for nearly spherical domains in the Bergman ball, extending known results from hyperbolic spaces to complex analysis settings.

Contribution

It proves the first isoperimetric inequality for nearly spherical sets in the Bergman ball, including a Fuglede theorem adaptation.

Findings

Proves a quantitative isoperimetric inequality in the Bergman ball.

Establishes the Fuglede theorem for nearly spherical sets.

First such result in the context of the Bergman ball.

Abstract

We prove a quantitative isoperimetric inequality for the nearly spherical subset of the Bergman ball in $\mathbb{C}^n$. We prove the Fuglede theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball and it makes the first result on the isoperimetric phenomenon in the Bergman ball.