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

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.