Toda lattice and Riemann type minimal surfaces

TL;DR

This paper explores the connection between Toda lattice and minimal surfaces, using integrable systems to construct new singly periodic minimal surfaces with higher genus, extending Riemann minimal surfaces.

Contribution

It introduces a novel method to construct higher-genus minimal surfaces via Toda lattice solutions, expanding the class of known minimal surfaces.

Findings

Constructed new singly periodic minimal surfaces with genus j(j+1)/2-1

Established a link between Toda lattice solutions and minimal surface configurations

Generalized Riemann minimal surfaces to higher genus cases

Abstract

Toda lattice and minimal surfaces are related to each other through Allen-Cahn equation. In view of the structure of the solutions of the Toda lattice, we find new balancing configuration using techniques of integrable systems. This allows us to construct new singly periodic minimal surfaces. The genus of these minimal surfaces equals $j(j+1)/2-1$. They are natural generalization of the Riemann minimal surfaces, which have genus zero.