18 apr 2017 -- 16:30
Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
Let $\Omega$ be a smooth bounded domain in $\mathbb{C}^n$. Z. Balogh and M. Bonk proved that if $\Omega$ is strictly pseudoconvex, then it must be Gromov-hyperbolic when endowed with the Kobayashi distance. After partial results by other authors, A. Zimmer proved in 2015 that a convex smooth domain in $\mathbb{C}^n$ is Gromov-hyperbolic if and only if it is of finite type, i.e. its complex tangent hyperplane at each point of the boundary has finite order of contact with it.
The question remains open in the class of pseudo-convex domains. We give a proof strategy, relying on the close relationship between Gromov hyperbolic spaces and metric trees, that aims at proving that if $n=2$ and $\Omega$ is $\mathbb{C}$-convex of finite type, then it should be Gromov-hyperbolic when endowed with the Kobayashi distance.