Euclidean minimal tori with planar ends and elliptic solitons

TL;DR

This paper characterizes Euclidean minimal tori with planar ends as a special class of Willmore tori with reducible spectral curves, linking them to elliptic KP solitons and Lamé potentials, and explores related examples like Riemann's staircase surfaces.

Contribution

It provides a spectral theoretic classification of minimal tori with planar ends and connects them to elliptic KP solitons and Lamé potentials, expanding understanding of their geometric and integrable structures.

Findings

Spectral characterization of Willmore tori with reducible spectral curves.

Connection between minimal tori with planar ends and elliptic KP solitons.

Examples include tori related to 1-gap Lamé potentials and Riemann's staircase surfaces.

Abstract

A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type is necessarily the double of the spectral curve of an elliptic KP soliton. The simplest possible examples of minimal tori with planar ends are related to 1--gap Lam\'e potentials, the simplest non--trivial algebro geometric KdV potentials. If one allows for translational periods, Riemann's "staircase" minimal surfaces appear as other examples related to 1--gap Lam\'e potentials.