Parametric inequalities and Weyl law for the volume spectrum

Guth, Larry; Liokumovich, Yevgeny

doi:10.2140/gt.2025.29.863

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Parametric inequalities and Weyl law for the volume spectrum

Larry Guth and Yevgeny Liokumovich

Geometry & Topology 29 (2025) 863–902

DOI: 10.2140/gt.2025.29.863

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: 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.

Keywords

isoperimetric inequality, coarea inequality, min-max theory, Weyl law, minimal surface

Mathematical Subject Classification

Primary: 35P20, 53A10, 53C23

References

Publication

Received: 3 March 2022

Revised: 16 September 2022

Accepted: 18 October 2022

Published: 21 April 2025

Proposed: John Lott

Seconded: Dmitri Burago, David Fisher

Authors

Larry Guth
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

Yevgeny Liokumovich
Department of Mathematics University of Toronto Toronto, ON Canada

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Volume 29, issue 2 (2025)