Minimal tori in $\mathbb{R}^4$

TL;DR

This paper explores minimal tori in four-dimensional space, using classical and new tools, and finds explicit solutions for certain tori, revealing existence and uniqueness results for minimal surfaces with specific curvature and topology.

Contribution

It introduces a system of equations characterizing minimal tori in , providing explicit solutions for rectangular tori and classifying solutions for square and equianharmonic tori.

Findings

Explicit solutions for rectangular tori.

Unique family of solutions for square tori.

No solutions for equianharmonic tori.

Abstract

We describe tools for the study of minimal surfaces in $\mathbb{R}^4$; some are classical (the Gauss maps) and some are newer (the link/braid/writhe at infinity). Then we look for complete proper non holomorphic minimal tori with total curvature $-8\pi$ and a single end immersed in $\mathbb{R}^4$. We translate the problem into a system of $10$ quadratic or linear equations in $11$ real variables with coefficients in terms of the Weierstrass function $\wp$ and give explicit solutions for these equations if $T$ is a rectangular torus. For the square torus, we have a complete answer with a unique family of solutions generalizing the Chen-Gackstetter torus in $\mathbb{R}^3$. On the other hand, we show that there is no solution on the equianharmonic torus.