Weighted $\infty$-Willmore Spheres

TL;DR

This paper studies the minimization of the weighted $L^ty$ norm of mean curvature on spheres, deriving a PDE system that describes the geometric behavior of solutions and their curvature properties.

Contribution

It introduces a second-order PDE system as the limit of Euler-Lagrange equations for weighted mean curvature minimization, revealing curvature constraints.

Findings

Solutions have mean curvature taking at most three values.

The PDE system constrains the geometric behavior of the surfaces.

Mean curvature is zero on the nodal set if it exists.

Abstract

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