The p-widths of a surface

Chodosh, Otis; Mantoulidis, Christos

doi:10.1007/s10240-023-00141-7

Article

The p-widths of a surface

Chodosh, Otis ; Mantoulidis, Christos

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

Résumé

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.

Reçu le : 2021-08-24
Accepté le : 2023-04-13
Première publication : 2023-05-04
Publié le : 2023-06-28

Zbl

DOI : 10.1007/s10240-023-00141-7

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

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