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Riemann-Hilbert problems with singularities

Banff International Research Station for Mathematical Innovation and Discovery

The study of analytic discs attached to a totally real submanifold M of $\mathbb C^n$ leads to the consideration of a regular Riemann-Hilbert problem of a special form. Following this approach, Forstneric, and later on Globevnik, characterized the existence and dimension of a family of deformations of a given analytic disc attached to M in terms of certain indices. However, in case M admits some complex tangencies, the indices mentioned above are no longer well-defined and the Forstneric-Globevnik method falls apart. In this talk, I will focus on a class of such singular Riemann-Hilbert problems. We will see that they can be solved by a factorization technique that reduces them to regular Riemann-Hilbert problems with geometric constraints. In particular, we will deduce the existence of stationary type discs attached to finite type hypersurfaces.

Mathematics; Several complex variables and analytic spaces; Potential theory; Global analysis

video/mp4

Author affiliation: American University of Beirut

2016-11-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59651

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/