Comparison of invariant functions on strongly pseudoconvex domains

TL;DR

This paper demonstrates that on strongly pseudoconvex domains, the Carathéodory distance and Lempert function are nearly identical, and with smooth boundaries, they closely relate to the Bergman distance, revealing deep connections among invariant metrics.

Contribution

The paper establishes near equivalences among key invariant functions on strongly pseudoconvex domains, extending understanding of their geometric relationships.

Findings

Carathéodory distance and Lempert function are almost the same.

With $C^{2+ ext{ε}}$ boundary smoothness, they relate closely to the Bergman distance.

The Bergman distance is approximately $rac{1}{ oot{n+1}}$ times the other metrics.

Abstract

It is shown that the Carath\'eodory distance and the Lempert function are almost the same on any strongly pseudoconvex domain in $\C^n;$ in addition, if the boundary is $C^{2+\eps}$-smooth, then $\sqrt{n+1}$ times one of them almost coincides with the Bergman distance.