Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

TL;DR

This paper investigates the geometric properties and Morse index of inverted minimal surfaces with finite total curvature, revealing a precise relationship between the number of ends and the Willmore index.

Contribution

It establishes an exact formula for the Willmore Morse index of inverted minimal spheres and projective planes with multiple ends, linking geometric configuration to stability.

Findings

The Willmore index equals the number of ends minus three.

Inverted minimal surfaces with all ends meeting at a point have specific stability properties.

The paper connects geometric configurations of minimal surfaces to their Willmore energy stability.

Abstract

We study complete minimal surfaces in $\mathbb{R}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy $\mathcal{W}: =\frac{1}{4} \int|\vec H|^2$. In codimension one, we prove that the $\mathcal{W}$-Morse index for any inverted minimal sphere or real projective plane with $m$ such ends is exactly $m-3=\frac{\mathcal{W}}{4\pi}-3$. We also consider several geometric properties -- for example, the property that all $m$ asymptotic planes meet at a single point -- of these minimal surfaces and explore their relation to the $\mathcal{W}$-Morse index of their inverted surfaces.