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A scattering theory of harmonic one-forms on Riemann surfaces
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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.
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Cited by 2 Pith papers
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Scattering theory on Riemann surfaces II: The scattering matrix and generalized period mappings
A unitary scattering matrix for harmonic one-forms on Riemann surfaces is constructed from Schiffer operators, and its associated polarizations unify the classical and universal Teichmüller period maps.
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Scattering theory on Riemann surfaces I: Schiffer operators, cohomology, and index theorems
For a compact Riemann surface split by quasicircles into connected pieces of genuses g1 and g2, the index of the Schiffer operator T_{1,2} equals g1 minus g2.
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