Parametric inequalities and Weyl law for the volume spectrum

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: the isoperimetric inequality and the 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.35P20, 53A10, 53C23 1. Introduction 863 2. Approximation results for families of cycles 869 3. Homotopy classes of the space of cycles 883 4. Parametric coarea inequality 888 5. Parametric isoperimetric inequality 895 6. Proof of the Weyl law 899