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 with $m$ such ends is exactly $m-3=\frac{\mathcal{W}}{4\pi}-3$, completing previous work. We 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.Comment: Comments welcom

A posteriori error estimates for wave maps into spheres

We provide a posteriori error estimates in the energy norm for temporal semi-discretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak-strong stability estimates which we combine with suitable reconstructions of the numerical solution. Numerical experiments are presented that confirm that our error estimates are formally optimal until the solution develops singularities