The p-widths of a surface

TL;DR

This paper investigates the $p$-widths of closed Riemannian surfaces, showing they correspond to unions of geodesics, and determines the $p$-widths of the round sphere, establishing a universal constant in a related law.

Contribution

It proves that $p$-widths of surfaces are unions of geodesics, calculates the $p$-widths of the round sphere, and introduces new theorems on stationary geodesic nets and their properties.

Findings

$p$-widths of surfaces are unions of geodesics.

The $p$-widths of the round sphere are attained by $loor{\sqrt{p} floor$ great circles.

Universal constant in the Liokumovich--Marques--Neves--Weyl law is $\sqrt{\pi}$.

Abstract

The $p$-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace--Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the $p$-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the $p$-widths of the round sphere are attained by $\lfloor \sqrt{p}\rfloor$ great circles. As a result, we find the universal constant in the Liokumovich--Marques--Neves--Weyl law for surfaces to be $\sqrt{\pi}$. En route to calculating the $p$-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik--Schnirelmann category zero.