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.

Off

2D hydrodynamics of ideal fluid with free surface is considered. A time-dependent conformal transformation is used which maps a free fluid surface into the real line with fluid domain mapped into the lower complex half-plane. The fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. The initially flat surface with the pole in the complex velocity turns over arbitrary small time into the branch cut connecting two square root branch points. Without gravity one of these branch points approaches the fluid surface with the approximate exponential law corresponding to the formation of the fluid jet. The addition of gravity results in wavebreaking in the form of plunging of the jet into the water surface. The infinite family of solutions with persistent poles in complex velocity is also found. These poles are generally coupled with branch points located at other points of the upper half-plane. Residues of these poles are new, previously unknown constants of motion. All these constants of motion commute with each other in the sense of underlying non-canonical Hamiltonian dynamics. It is suggested that the existence of these extra constants of motion provides an argument in support of the conjecture of complete integrability of 2D free surface hydrodynamics. These results are verified in details through high precision simulations (a variable precision up to 200 digits is used to reliable recover the structure of complex singularities). The use of the additional conformal transformation to resolve the dynamics near branch points allows to speed up simulations more that 10^8 times and observe a formation of multiple Crapper capillary solutions during overturning of the wave contributing to the turbulence of surface wave. The analytical structure of Stokes wave is also analyzed. For non-limiting Stokes wave the only singularity in the physical sheet of Riemann surface is the square-root branch point located. The corresponding branch cut defines the second sheet of the Riemann surface if one crosses the branch cut. The infinite number of pairs of square root singularities is found corresponding to infinite number of non-physical sheets of Riemann surface. Each pair belongs to its own non-physical sheet of Riemann surface. Increase of the steepness of the Stokes wave means that all these singularities simultaneously
approach the real line from different sheets of Riemann surface and merge together forming 2/3 power law singularity of the limiting Stokes wave. It is shown that non-limiting Stokes wave at the leading order consists of the infinite product of nested square root singularities which form the infinite number of sheets of Riemann surface.

Off