Parametric inequalities and Weyl law for the volume spectrum

Abstract

We show that the Weyl law for the volume spectrum in a compact Riemannian manifold conjectured by Gromov can be derived from parametric generalizations of two famous inequalities: isoperimetric inequality and coarea inequality. We prove two such generalizations in low dimensions and obtain the Weyl law for 1-cycles in 3-manifolds. We also give a new proof of the Almgren isomorphism theorem.

arXiv commentProof of Prop. 2.7 corrected. To appear in Geometry and Topology

References in corpus (3)

• Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
• A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory
• Families of short cycles on Riemannian surfaces