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

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.