A notion of visibility based on the Kobayashi distance, and applications

speaker: Gautam Bharali (Indian Institute of Science, Bangalore)

abstract: Given any 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 recent work work with Andrew Zimmer -- possess this visibility property. This is a rather broad class of domains that includes, for instance, all pseudoconvex domains of finite type. It also includes a range of convex domains for which it is known that the Kobayashi distance is \emph{not} Gromov hyperbolic. Among the phenomena that one can demonstrate for such visibility spaces is a form of the Wolff-Denjoy theorem. We shall give a proof of this result, focusing on the role of visibility in our proof. If time permits, we shall look at other applications and at domains that do not have the``Goldilocks'' properties but are visibility spaces in the above-mentioned sense. This talk presents joint work with Andrew Zimmer and with Anwoy Maitra.

timetable:
Wed 23 May, 11:40 - 12:30, Aula Dini
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