Seminar: Riccardo Ugolini (G.15.25)
The Kobayashi pseudodistance on a complex manifold is defined as an infimum over chains of holomorphic disks. A manifold is (Kobayashi) hyperbolic if this pseudodistance is non-degenerate and hence defines a genuine distance.
A complex contact manifold is a complex manifold endowed with a corank-one holomorphic distribution, in analogy with real contact geometry. Inspired by constructions in sub-Riemannian geometry, one may restrict attention to holomorphic disks tangent to the contact distribution and define a corresponding Kobayashi-type pseudodistance. A complex contact manifold is called contact hyperbolic when this pseudodistance is a true distance.
In this talk, we introduce the theory of Kobayashi hyperbolicity and explain how it extends naturally to the contact setting. As an application, we show that fixing the dimension of the group of complex contact automorphisms imposes strong structural constraints on the underlying manifold.
