The CAT(0) geometry of convex domains with the Kobayashi metrics

TL;DR

This paper investigates the geometric properties of convex domains in complex spaces with the Kobayashi metric, establishing conditions under which these domains exhibit CAT(0) geometry, with results depending on dimension and boundary smoothness.

Contribution

It proves that m-convexity is necessary for CAT(0) geometry in 2D convex domains and extends similar results to higher dimensions with boundary smoothness assumptions.

Findings

m-convexity is necessary for CAT(0) in 2D convex domains

Higher-dimensional results require boundary smoothness

Provides geometric criteria linking convexity and CAT(0) properties

Abstract

Let $(\Omega,K_{\Omega})$ be a convex domain in $\mathbb C^d$ with the Kobayashi metric $K_{\Omega}$. In this paper we prove that $m$-convexity is a necessary condition for $(\Omega, K_{\Omega})$ to be CAT(0) if $d=2$. Moreover, when $\Omega \subset \mathbb{C}^d, \: d\geq3$, we obtain a similar result with the further smoothness assumption on its boundary.