Weighted ∞-Willmore Spheres

On the two-sphere Σ, we consider the problem of minimising among suitable immersions f:Σ→R ³ the weighted L ^∞ norm of the mean curvature H, with weighting given by a prescribed ambient function ξ, 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→∞ 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∈{±‖ξH‖L _∞} away from the nodal set of the PDE system, and H=0 on the nodal set (if it is non-empty).