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arXiv · 2110.14367

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

Jonas Hirsch·Rob Kusner·Elena Mäder-Baumdicker
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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.

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Jonas Hirsch, Rob Kusner, Elena Mäder-Baumdicker. 2021-10-27. Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions. https://arxiv.org/abs/2110.14367

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