2021/12/01 by Eric Schippers, Wolfgang Staubach, Schippers, Eric +1 · 1 citation
Mathematics · #14F40 #30F15 #30F30 #35P99 #51M15 #Algebraic Geometry (math.AG) #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2112.00835
openalex publication_date 2021/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.