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

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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