On the local (non-)extendability of germs of holomorphic functions
Abstract: In one complex variable, the Riemann mapping theorem says that every simply connected, smoothly bounded domain D in C is biholomorphically equivalent to the unit disc. In particular, any such domain D is a domain of holomorphy; i.e., one can construct a holomorphic function on D that cannot be extended to any larger domain.
The geometry of domains in CN, N > 1, is rather complicated. As an example, the open ball and the polydisc in CN are domains of holomorphy; however, Poincar\'e showed in 1906 that they are not biholomorphic! At the same time, Hartogs constructed an example of a domain in C2 that is not a domain of holomorphy.
In this talk we will discuss some geometric properties of the boundaries of real domains in CN, N>1, and their relation to the (non-)extendability of holomorphic functions.