A link at infinity for minimal surfaces in $\mathbb{R}^4$

TL;DR

This paper introduces the concept of a link at infinity for complete minimal surfaces in four-dimensional space, providing tools to analyze their geometry and self-intersection properties.

Contribution

It extends the notion of link at infinity from complex curves to minimal surfaces in , deriving a formula for total normal curvature and applications to genus zero surfaces.

Findings

Derived the writhe number at infinity for minimal surfaces.

Provided a formula relating link at infinity to total normal curvature.

Illustrated how the link at infinity can determine self-intersection properties.

Abstract

We look at complete minimal surfaces of finite total curvature in $\mathbb{R}^4$. Similarly to the case of complex curves in $\mathbb{C}^2$ we introduce their {\it link at infinity}; we derive the {\it writhe number at infinity} which gives a formula for the total normal curvature of the surface. The knowledge of the link at infinity can sometimes help us determine if a surface has self-intersection and we illustrate this idea by looking at genus zero surfaces of small total curvature.