The \(p\)-widths \(\{\omega_p\}_{p=1}^\infty\) of a compact Riemannian manifold \(M=M^{n+1}\) can be viewed as one possible nonlinear counterpart of the well-known spectrum \(\{\lambda_p\}_{p=1}^\infty\) of the Laplace-Beltrami operator on \(M\). The analogy rests on the minimax characterisation, using Rayleigh quotients, of the eigenvalues of the Laplace-Beltrami operator. In the case of \(p\)-widths, a notion of area (for submanifolds of \(M\)) replaces the Dirichlet energy. Of particular relevance is the case of \(n\)-area (for hypersurfaces, that is, submanifolds of codimension \(1\)). The idea of \(p\)-widths was first introduced in [Lect. Notes Math. 1317, 132–184 (1988; Zbl 0664.41019)] by M. Gromov, and was studied by Gromov himself, and by L. Guth [Geom. Funct. Anal. 18, No. 6, 1917–1987 (2009; Zbl 1190.53038)]. In more recent years, \(p\)-widths have entered minimal surface theory with a decisive role in combination with (Almgren-Pitts) minmax constructions. In analogy with the classical Weyl law for the Laplace spectrum \(\{\lambda_p\}_{p=1}^\infty\), Y. Liokumovich et al. [Ann. Math. (2) 187, No. 3, 933–961 (2018; Zbl 1390.53034)] proved a Weyl law for the “volume spectrum” \(\{\omega_p\}_{p=1}^\infty\): \[ \lim_{p\to \infty} \omega_p \, p^{-\frac{1}{n+1}}= a(n) \text{Vol}(M)^{\frac{n}{n+1}}, \] for a dimensional constant \(a(n)\). While this was sufficient for striking applications to the existence of minimal hypersurfaces (e.g. [K. Irie et al., Ann. Math. (2) 187, No. 3, 963–972 (2018; Zbl 1387.53083)]), the explicit value of the constant \(a(n)\) had remained unknown, and the values of \(\omega_p\) had remained unknown even for the simplest choices of \(M\).
The present paper focuses on the case \(n=1\) (\(M\) is a closed surface) and analyses multi-parameter minmax constructions. It was known that \(\omega_p\) appears as the minmax value for the construction with \(p\) parameters, however, the geometric object produced by the minmax was only characterised as a geodesic net and \(\omega_p\) corresponds to the length of this net (counting possible integer-valued multiplicities). The first result in the present paper establishes that the geodesic net obtained from the minmax construction is actually a union of (connected) closed immersed geodesics, each appearing with integer-valued multiplicity. This greatly extends previous results, obtained by E. Calabi and J. Cao for \(p=1\) [J. Differ. Geom. 36, No. 3, 517–549 (1992; Zbl 0768.53019)] and by N. Sarquis Aiex for \(p\in \{1, 2, \dots, 8\}\) [Commun. Anal. Geom. 27, No. 2, 251–285 (2019; Zbl 1440.53043)]. The approach pursued in the paper uses a phase-transition regularization of the area-functional, commonly referred to as Allen-Cahn minmax. (Thanks to a result by A. Dey [Geom. Funct. Anal. 32, No. 5, 980–1040 (2022; Zbl 1515.53009)], the \(p\)-widths obtained via the Almgren-Pitts and Allen-Cahn minmax constructions are the same.) The choice of the potential in the present paper is, for analytic reasons, quite specific: it is based on the elliptic sine-Gordon equation.
The second main result is a computation of the full \(p\)-width spectrum of \(S^2\), the two-sphere with the standard round metric: \[ \omega_p(S^2) = 2 \pi \lfloor \sqrt{p} \rfloor. \] The corresponding geometric object is a union of \(\lfloor \sqrt{p} \rfloor\) great circles (possibly repeated). A corollary of this is the explicit determination of \(a(1)\) in the above Weyl law, \[ a(1)=\sqrt{\pi}. \] This is the first result of this kind for the nonlinear volume spectrum.