Title: Two Bergman-type interpolation problems on finite Riemann surfaces
Speaker: Dror Varolin
Speaker Info: Stony Brook
Special Note: Unusual time and location
Let X be an open Riemann surface with a Hermitian metric and a weight function (i.e. non-trivial metric for the trivial line bundle). Given a closed discrete subset G in X, the above data defines a Bergman space on X and a Hilbert space on G (in a standard way). We say that G is an interpolation set if the restriction map from the Bergman space on X to the Hilbert space on G is surjective. The interpolation problem consists in characterizing all interpolation sets.Date: Thursday, April 16, 2015
When X is the complement of a finite set in a compact Riemann surface (i.e., a compact Riemann surface with some finite number of punctures), the metric g is flat outside some compact subset of X, and the curvature of the weight satisfies certain positivity and boundedness conditions, we give a complete solution to the interpolation problem.
We then turn our attention to more general bordered Riemann surfaces with finitely many punctures. We equip these with the unique metric of constant negative curvature -4, and point out that the same Bergman interpolation problem discussed above does not have a reasonable solution. We therefore modify the problem so that it doesn't change in the asymptotically flat case, but has a reasonable solution in the hyperbolic case. Finally, we give a complete characterization of interpolation sets for this modified problem.