Parametric inequalities and Weyl law for the volume spectrum

TL;DR

This paper derives the Weyl law for the volume spectrum in compact Riemannian manifolds using parametric inequalities, providing new proofs and extending results to 1-cycles in 3-manifolds.

Contribution

It introduces parametric generalizations of isoperimetric and coarea inequalities, leading to the Weyl law for 1-cycles and a new proof of the Almgren isomorphism theorem.

Findings

Weyl law for 1-cycles in 3-manifolds established

Parametric inequalities proved in low dimensions

New proof of Almgren isomorphism theorem provided

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.