Families of strictly pseudoconvex domains and peak functions

TL;DR

This paper demonstrates that for a smoothly varying family of strictly pseudoconvex domains, there exists a corresponding family of peak functions that vary continuously with the domains and boundary points.

Contribution

It establishes the existence of a continuously varying family of peak functions for smoothly changing strictly pseudoconvex domains.

Findings

Existence of continuous family of peak functions for all domains in the family.

Peak functions vary continuously with domain and boundary point.

Applicable to families of domains with $ ext{C}^2$ topology.

Abstract

We prove that given a family $(G_t)$ of strictly pseudoconvex domains varying in $\mathcal{C}^2$ topology on domains, there exists a continuously varying family of peak functions $h_{t,\zeta}$ for all $G_t$ at every $\zeta\in\partial G_t.$