The p-widths of a surface

Otis Chodosh; Christos Mantoulidis

doi:10.1007/s10240-023-00141-7

Article

The p-widths of a surface

Otis Chodosh ¹; Christos Mantoulidis ¹

¹

Publications Mathématiques de l'IHÉS, Volume 137 (2023), pp. 245-342

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 $⌊ \sqrt{p} ⌋$ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be $\sqrt{π}$ .

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.

Received: 2021-08-24
Accepted: 2023-04-13
Online First: 2023-05-04
Published online: 2023-06-07

Zbl

DOI: 10.1007/s10240-023-00141-7

Author's affiliations:

Otis Chodosh ¹; Christos Mantoulidis ¹

¹

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     author = {Otis Chodosh and Christos Mantoulidis},
     title = {The p-widths of a surface},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {245--342},
     year = {2023},
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Otis Chodosh; Christos Mantoulidis. The p-widths of a surface. Publications Mathématiques de l'IHÉS, Volume 137 (2023), pp. 245-342. doi: 10.1007/s10240-023-00141-7

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