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

We study complete minimal surfaces in Rn with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy W:=1/4∫ |H→|². In codimension one, we prove that the W-Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly m-3 =W/4π-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 W-Morse index of their inverted surfaces.