Scattering theory on Riemann surfaces II: The scattering matrix and generalized period mappings

TL;DR

This paper develops a scattering theory for harmonic one-forms on Riemann surfaces, providing explicit formulas for the scattering matrix, establishing its unitarity, and linking it to generalized period mappings and polarizations in Teichmüller theory.

Contribution

It introduces a new scattering framework for harmonic forms on Riemann surfaces, connecting boundary value problems with integral operators and extending classical polarization concepts.

Findings

Explicit expression for the scattering matrix in terms of Schiffer operators

Proof that the scattering matrix is unitary

Establishment of a connection between polarizations and the universal Teichmüller space

Abstract

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems involving systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of integral operators which we call Schiffer operators, and show that the matrix is unitary. We also obtain a general association of positive polarizing Lagrangian spaces to bordered Riemann surfaces, which unifies the classical polarizations for compact surfaces of algebraic geometry with the infinite-dimensional period map of the universal Teichm\"uller space.