Weighted $\mathrm{\infty }$-Willmore Spheres

On the two-sphere $\mathrm{\Sigma }$, we consider the problem of minimising among suitable immersions $f\text{\hspace{0.17em}⁣}:\mathrm{\Sigma }\to {\mathbb{R}}^3$ the weighted $L^{\mathrm{\infty }}$ norm of the mean curvature $H$, with weighting given by a prescribed ambient function $\xi$, subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of ``pseudo-minimiser’’ surfaces must satisfy a second-order PDE system obtained as the limit as $p\to \mathrm{\infty }$ of the Euler-Lagrange equations for the approximating $L^p$ problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: $H\in \{\pm \mathrm{\mid }\mathrm{\mid }\xi H\mathrm{\mid }{\mathrm{\mid }}_{L^{\mathrm{\infty }}}\}$ away from the nodal set of the PDE system, and $H=0$ on the nodal set (if it is non-empty).