Weighted ∞-Willmore spheres

On the two-sphere Σ Σ , we consider the problem of minimising among suitable immersions f \,:Σ → R³ f : Σ → R 3 the weighted L^∞ L ∞ norm of the mean curvature H , with weighting given by a prescribed ambient function ξ 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 → ∞ p → ∞ of the Euler–Lagrange equations for the approximating Lᵖ 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 ∈ \ ± ξ H L^∞ \ H ∈ { ± ‖ ξ H ‖ L ∞ } away from the nodal set of the PDE system, and $$H = 0$$ H = 0 on the nodal set (if it is non-empty).