15 oct 2019 -- 16:00
Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
Given a metric space, there are several notions of it being negatively curved. In this talk, we single out a weak notion of negative curvature (which, in fact, is a consequence of negative curvature in the Riemannian category) that turns out to be very useful in proving results about holomorphic maps. This property is a form of visibility, the underlying metric spaces being bounded domains in $\mathbb{C}^n$ equipped with the Kobayashi distance. In this talk, we shall present a general quantitative condition for a domain to be a visibility space in the sense alluded to above. A class of domains known as Goldilocks domains -- introduced in joint work with Andrew Zimmer in 2017 -- possess this visibility property. Visibility domains form a broad class of domains that includes, for instance, all pseudoconvex domains of finite type. Throughout the talk, we shall refer to the Wolff-Denjoy theorem -- which was previously known to hold true on certain convex domains and on strongly pseudoconvex domains -- as a framing device for the sort of phenomena that extend to visibility domains. We shall also discuss methods for determining when a domain is a visibility space and for constructing new examples with rough boundaries. This is joint work with Andrew Zimmer and Anwoy Maitra.