Harmonic tori in de Sitter spaces $S^{2n}_1$

TL;DR

This paper demonstrates that superconformal harmonic immersions from genus one surfaces into de Sitter spaces are of finite type, enabling their construction through solving ordinary differential equations, with applications to Willmore tori in 3-spheres.

Contribution

It establishes that all such harmonic immersions are of finite type and provides a method to construct Willmore tori in $S^3$ without umbilic points.

Findings

Superconformal harmonic immersions are of finite type.

All Willmore tori in $S^3$ without umbilic points can be constructed via ODEs.

Abstract

We show that all superconformal harmonic immersions from genus one surfaces into de Sitter spaces $ S ^ {2n}_1 $ with globally defined harmonic sequence are of finite-type and hence result merely from solving a pair of ordinary differential equations. As an application, we prove that all Willmore tori in $ S ^ 3 $ without umbilic points can be constructed in this simple way.