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 $\Omega \subset {\mathbb{C}}^d,d\ge 3$, we obtain a similar result with the further smoothness assumption on its boundary.