The p-widths of a surface

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.

arXiv commentFinal version, to appear in Publ. Math. Inst. Hautes Études Sci

References in corpus (13)

• Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
• Minimal hypersurfaces with bounded index
• A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory
• Minimal two-spheres in three-spheres
• A proof of the multiplicity one conjecture for min-max minimal surfaces in arbitrary codimension
• The index and nullity of the Lawson surfaces ${\xi }_{g,1}$
• Allen-Cahn min-max on surfaces
• On the existence of minimal Heegaard surfaces
• Families of short cycles on Riemannian surfaces
• Orientability of min-max hypersurfaces in manifolds of positive Ricci curvature
• Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities
• Generic Regularity of Minimal Hypersurfaces in Dimension 8
• Ground states of semilinear elliptic equations

Cited by in corpus (1)

• The first $p$-widths of the unit disk