Chodosh, OtisMantoulidis, Christos2023-07-212023-07-212023Chodosh, Otis and Mantoulidis, Christos. "The p-widths of a surface." <i>Publications mathématiques de l'IHÉS,</i> 137, (2023) Springer Nature: 245-342. https://doi.org/10.1007/s10240-023-00141-7.https://hdl.handle.net/1911/115010The 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 ⌊p–√⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be 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.engJournal articles10240-023-00141-7https://doi.org/10.1007/s10240-023-00141-7