SQUARE-INTEGRABLE DIFFERENTIALS ON OPEN RIEMANN SURFACES

Meromorphic Differentials with Twisted Coefficients on Compact Riemann Surfaces

This note is to concern a generalization to the case of twisted coefficients of the classical theory of Abelian differentials on a compact Riemann surface. We apply the Dirichlet’s principle to a modified energy functional to show the existence of differentials with twisted coefficients of the second and third kinds under a suitable assumption on residues. 1 Main results and discussion Let X be...

Non - Abelian Vortices on Riemann Surfaces : an Integrable Case ∗

We consider U(n+1) Yang-Mills instantons on the space Σ×S2, where Σ is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n+1) instanton equations on Σ× S are equivalent to non-Abelian vortex equations on Σ. Solutions to these equations are given by pairs (A, φ), where A is a gauge potential of the group U(n) and φ is a Higgs field in the ...

Ideal Theory on Open Riemann Surfaces

Introduction. The theorems of the classical ideal theory in fields of algebraic numbers hold in rings of analytic functions on compact Riemann surfaces. The surfaces admitted in our discussion are closely related to algebraic surfaces; we deal either with compact surfaces from which a finite number of points are omitted or, more generally, with surfaces determined by an algebroid function. The ...

Integrable Systems and Riemann Surfaces Lecture Notes (preliminary version)

1 KdV equation and Schrödinger operator 2 1.1 Integrability of Korteweg – de Vries equation . . . . . . . . . . . . . . . . . . 2 1.2 Elements of scattering theory for the Schrödinger operator . . . . . . . . . . . 5 1.3 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Dressing operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Integrable systems and Riemann surfaces of infinite genus