Parametric inequalities and Weyl law for the volume spectrum

Guth, Larry¹ ; Liokumovich, Yevgeny²

¹Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States
²Department of Mathematics, University of Toronto, Toronto, ON, Canada

Geometry & topology, Tome 29 (2025) no. 2, pp. 863-902

Résumé

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.

Détail

DOI : 10.2140/gt.2025.29.863

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

Bibliographie

[1] F J Almgren Jr., The homotopy groups of the integral cycle groups, Topology 1 (1962) 257 | DOI

[2] F J Almgren Jr., The theory of varifolds, mimeographed notes (1964)

[3] O Chodosh, C Mantoulidis, Minimal surfaces and the Allen–Cahn equation on 3-manifolds : index, multiplicity, and curvature estimates, Ann. of Math. 191 (2020) 213 | DOI

[4] O Chodosh, C Mantoulidis, The p-widths of a surface, Publ. Math. Inst. Hautes Études Sci. 137 (2023) 245 | DOI

[5] A Dey, A comparison of the Almgren–Pitts and the Allen–Cahn min-max theory, Geom. Funct. Anal. 32 (2022) 980 | DOI

[6] H Federer, Geometric measure theory, 153, Springer (1969) | DOI

[7] W H Fleming, Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966) 160 | DOI

[8] P Gaspar, M A M Guaraco, The Weyl law for the phase transition spectrum and density of limit interfaces, Geom. Funct. Anal. 29 (2019) 382 | DOI

[9] P Glynn-Adey, Y Liokumovich, Width, Ricci curvature, and minimal hypersurfaces, J. Differential Geom. 105 (2017) 33 | DOI

[10] M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1

[11] M Gromov, Dimension, nonlinear spectra and width, from: "Geometric aspects of functional analysis" (editors J Lindenstrauss, V D Milman), Lecture Notes in Math. 1317, Springer (1988) 132 | DOI

[12] M Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003) 178 | DOI

[13] M Gromov, Singularities, expanders and topology of maps, I : Homology versus volume in the spaces of cycles, Geom. Funct. Anal. 19 (2009) 743 | DOI

[14] L Guth, The width-volume inequality, Geom. Funct. Anal. 17 (2007) 1139 | DOI

[15] L Guth, Minimax problems related to cup powers and Steenrod squares, Geom. Funct. Anal. 18 (2009) 1917 | DOI

[16] K Irie, F C Marques, A Neves, Density of minimal hypersurfaces for generic metrics, Ann. of Math. 187 (2018) 963 | DOI

[17] X Li, B Staffa, On the equidistribution of closed geodesics and geodesic nets, Trans. Amer. Math. Soc. 376 (2023) 8825 | DOI

[18] Y Li, Existence of infinitely many minimal hypersurfaces in higher-dimensional closed manifolds with generic metrics, J. Differential Geom. 124 (2023) 381 | DOI

[19] Y Liokumovich, Families of short cycles on Riemannian surfaces, Duke Math. J. 165 (2016) 1363 | DOI

[20] Y Liokumovich, B Lishak, A Nabutovsky, R Rotman, Filling metric spaces, Duke Math. J. 171 (2022) 595 | DOI

[21] Y Liokumovich, F C Marques, A Neves, Weyl law for the volume spectrum, Ann. of Math. 187 (2018) 933 | DOI

[22] Y Liokumovich, B Staffa, Generic density of geodesic nets, Selecta Math. 30 (2024) 14 | DOI

[23] F C Marques, A Neves, Min-max theory and the Willmore conjecture, Ann. of Math. 179 (2014) 683 | DOI

[24] F C Marques, A Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math. 209 (2017) 577 | DOI

[25] F C Marques, A Neves, Morse index of multiplicity one min-max minimal hypersurfaces, Adv. Math. 378 (2021) 107527 | DOI

[26] F C Marques, A Neves, A Song, Equidistribution of minimal hypersurfaces for generic metrics, Invent. Math. 216 (2019) 421 | DOI

[27] L Mazurowski, A Weyl law for the p-Laplacian, J. Funct. Anal. 288 (2025) 110734 | DOI

[28] J T Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, 27, Princeton Univ. Press (1981) | DOI

[29] A Song, A dichotomy for minimal hypersurfaces in manifolds thick at infinity, Ann. Sci. École Norm. Sup. 56 (2023) 1085 | DOI

[30] A Song, Existence of infinitely many minimal hypersurfaces in closed manifolds, Ann. of Math. 197 (2023) 859 | DOI

[31] A Song, X Zhou, Generic scarring for minimal hypersurfaces along stable hypersurfaces, Geom. Funct. Anal. 31 (2021) 948 | DOI

[32] B Staffa, Bumpy metrics theorem for geodesic nets, J. Topol. Anal. (2023) | DOI

[33] B Staffa, Parametric coarea inequality for 1-cycles, preprint (2024)

[34] B Staffa, Weyl law for 1-cycles, preprint (2024)

[35] S Wenger, A short proof of Gromov’s filling inequality, Proc. Amer. Math. Soc. 136 (2008) 2937 | DOI

[36] R E Williamson Jr., Cobordism of combinatorial manifolds, Ann. of Math. 83 (1966) 1 | DOI

[37] X Zhou, On the multiplicity one conjecture in min-max theory, Ann. of Math. 192 (2020) 767 | DOI