A scattering theory of harmonic one-forms on Riemann surfaces

TL;DR

This paper develops a scattering theory for harmonic one-forms on Riemann surfaces, providing explicit formulas for the scattering matrix, proving its unitarity, and linking integral operators to topological invariants and Teichmüller theory.

Contribution

It introduces a novel scattering framework for harmonic forms on Riemann surfaces, connecting boundary value problems with topological and geometric invariants.

Findings

Explicit expression for the scattering matrix in terms of Schiffer operators

Proof that the scattering matrix is unitary

Establishment of index theorems relating integral operators to topological invariants

Abstract

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems through 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. As a consequence of this scattering theory, we prove index theorems relating these conformally invariant integral operators to topological invariants. 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 Teichmueller space.