Euclidean Minimal Tori with Planar Ends and Elliptic Solitons

Abstract

A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3-sphere |$S^3=\mathbb {R}^3\cup \{\infty \}$|⁠. 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 Kadomtsev-Petviashvili (KP) soliton. The simplest possible examples of minimal tori with planar ends are related to 1-gap Lamé potentials, the simplest nontrivial algebro geometric Korteweg-de Vries (KdV) potentials. If one allows for translational periods, Riemann's “staircase” minimal surfaces appear as other examples related to 1-gap Lamé potentials.