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

Abstract

AbstractWe study complete minimal surfaces in $$\mathbb {R}^n$$ 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 $$\mathcal {W}: =\frac{1}{4} \int |\vec H|^2$$ W : = 1 4 ∫ | H → | 2 . In codimension one, we prove that the $$\mathcal {W}$$ 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 = W 4 π - 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 $$\mathcal {W}$$ W -Morse index of their inverted surfaces.