Nonlinear Waves Seminar - Chaya Norton

The behavior of differentials with real periods under degeneration of the Riemann surface

Given a Riemann surface with marked points and prescribed singular parts at those points, there exists a unique differential with such singularities and real periods. These differentials will be called RN differentials. The imaginary part of the abelian integral of an RN differential is a harmonic function on the Riemann surface, and similarly a harmonic function defines an RN differential. In the form of its harmonic function, RN differentials with simple poles were studied by Maxwell, as the potentials of the electromagnetic field created by point charges.

RN differentials play a role in multiple areas. For example in the 80's Krichever introduced RN differentials in the development of the spectral theory of the non-stationary Schrodinger operator. RN differentials on hyper-elliptic Riemann surfaces appear when applying the nonlinear steepest descent method for the Riemann-Hilbert Problem arising in the semiclassical limit of solutions to the one-dimensional focusing Nonlinear Schrodinger equation.

This talk will focus on using RN differentials as a tool to understand the geometry of the moduli space of genus g Riemann surfaces. Specifically because RN differentials are unique upon prescribing its singular structure, a finite bundle over the moduli space of Riemann surfaces given by the data of marked points and singular parts exhibits a section into the bundle of meromorphic differentials. We study how the section extends over the boundary of the Deligne-Mumford compactification of the moduli space of Riemann surfaces.

The residues which appear at nodes in the limit as the Riemann surface degenerates are described by solving a Kirchhoff problem, thus the RN section extends in a blow-up compactification, i.e. the space where solutions to the corresponding Kirchhoff problems extends. This result was achieved by introducing a new technique, solving a parametric jump problem. Additionally this tool allows us to describe all possible limits of zeros of such differentials and show they are given explicitly as zeros of twisted RN differentials.

The talk will focus on the perspective RN differentials provides for studying the geometry of M_g as well as a discussion of our results describing how RN differentials degenerate. This is joint work with Samuel Grushevsky and Igor Krichever.