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

We study complete minimal surfaces in \(\mathbb {R}^n\) ${\mathbb{R}}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy \(\mathcal {W}: =\frac{1}{4} \int |\vec H|^2\) $\mathcal{W}:=\frac{1}{4}\int {|\overrightarrow{H}|}^2$ . In codimension one, we prove that the \(\mathcal {W}\) $\mathcal{W}$ -Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly \(m-3=\frac{\mathcal {W}}{4\pi }-3\) $m-3=\frac{\mathcal{W}}{4\pi }-3$ . 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 \(\mathcal {W}\) $\mathcal{W}$ -Morse index of their inverted surfaces.