Weighted $$\infty $$-Willmore spheres

Abstract

AbstractOn the two-sphere $$\Sigma $$ Σ , we consider the problem of minimising among suitable immersions $$f \,:\Sigma \rightarrow \mathbb {R}^3$$ f : Σ → R 3 the weighted $$L^\infty $$ L ∞ norm of the mean curvature H, with weighting given by a prescribed ambient function $$\xi $$ 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 $$p \rightarrow \infty $$ p → ∞ of the Euler–Lagrange equations for the approximating $$L^p$$ 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: $$H \in \{ \pm \Vert \xi H\Vert _{L^\infty } \}$$ 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).