Whitham Deformations and the Space of Harmonic Tori in $\mathbb{S}^3$

TL;DR

This paper explores the structure of harmonic maps from a 2-torus to the 3-sphere, revealing smoothness and dimensional properties using spectral curves and Whitham deformations, and analyzing minimal tori singularities.

Contribution

It introduces a detailed analysis of the harmonic map space via spectral curve methods and characterizes the smooth and singular points related to minimal tori.

Findings

The harmonic map space is smooth and two-dimensional in a dense subset.

Minimal tori points are either smooth of dimension two or singular.

The spectral curve and Whitham deformation techniques are effective in this analysis.

Abstract

In this paper we investigate the space of harmonic maps from a 2-torus to $\mathbb{S}^3$ using the spectral curve correspondence and Whitham deformations. In an open and dense subset of a parameter space we find that the space of harmonic maps is smooth and has dimension two. We also show that the points that correspond to minimal tori (conformal harmonic maps) are either smooth points of dimension two or singular.