Spectral Curves for Cauchy–Riemann Operators on Punctured Elliptic Curves

We show that one can define a spectral curve for the Cauchy–Riemann operator on a punctured elliptic curve under appropriate boundary conditions. The algebraic curves thus obtained arise, for example, as irreducible components of the spectral curves of minimal tori with planar ends in $\mathbb{R}^3$. It turns out that these curves coincide with the spectral curves of certain elliptic KP solitons studied by Krichever.