Spectral curves for Cauchy-Riemann operators on punctured elliptic curves

TL;DR

This paper constructs spectral curves for Cauchy-Riemann operators on punctured elliptic curves with specific boundary conditions, linking them to spectral curves of minimal tori with planar ends and elliptic KP solitons.

Contribution

It introduces a method to define spectral curves for these operators, connecting geometric and integrable systems concepts.

Findings

Spectral curves correspond to irreducible components of minimal tori with planar ends.

These spectral curves match those of certain elliptic KP solitons.

The approach bridges complex analysis, algebraic geometry, and integrable systems.

Abstract

We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves coincide with the spectral curves of certain elliptic KP solitons as studied by Krichever.