Weighted $$\infty $$ -Willmore spheres

On the two-sphere \(\Sigma \), we consider the problem of minimising among suitable immersions \(f \,:\Sigma \rightarrow \mathbb {R}^3\) the weighted \(L^\infty \) 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 \(p \rightarrow \infty \) 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: \(H \in \{ \pm \Vert \xi H\Vert _{L^\infty } \}\) away from the nodal set of the PDE system, and \(H = 0\) on the nodal set (if it is non-empty).