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Parametric inequalities and Weyl law for the volume spectrum

Larry Guth and Yevgeny Liokumovich

Geometry & Topology 29 (2025) 863–902
Bibliography
1 F J Almgren Jr., The homotopy groups of the integral cycle groups, Topology 1 (1962) 257 MR0146835
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 MR4045964
4 O Chodosh, C Mantoulidis, The p-widths of a surface, Publ. Math. Inst. Hautes Études Sci. 137 (2023) 245 MR4588597
5 A Dey, A comparison of the Almgren–Pitts and the Allen–Cahn min-max theory, Geom. Funct. Anal. 32 (2022) 980 MR4498838
6 H Federer, Geometric measure theory, 153, Springer (1969) MR0257325
7 W H Fleming, Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966) 160 MR0185084
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 MR3945835
9 P Glynn-Adey, Y Liokumovich, Width, Ricci curvature, and minimal hypersurfaces, J. Differential Geom. 105 (2017) 33 MR3592694
10 M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1 MR0697984
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 MR0950979
12 M Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003) 178 MR1978494
13 M Gromov, Singularities, expanders and topology of maps, I : Homology versus volume in the spaces of cycles, Geom. Funct. Anal. 19 (2009) 743 MR2563769
14 L Guth, The width-volume inequality, Geom. Funct. Anal. 17 (2007) 1139 MR2373013
15 L Guth, Minimax problems related to cup powers and Steenrod squares, Geom. Funct. Anal. 18 (2009) 1917 MR2491695
16 K Irie, F C Marques, A Neves, Density of minimal hypersurfaces for generic metrics, Ann. of Math. 187 (2018) 963 MR3779962
17 X Li, B Staffa, On the equidistribution of closed geodesics and geodesic nets, Trans. Amer. Math. Soc. 376 (2023) 8825 MR4669312
18 Y Li, Existence of infinitely many minimal hypersurfaces in higher-dimensional closed manifolds with generic metrics, J. Differential Geom. 124 (2023) 381 MR4602728
19 Y Liokumovich, Families of short cycles on Riemannian surfaces, Duke Math. J. 165 (2016) 1363 MR3498868
20 Y Liokumovich, B Lishak, A Nabutovsky, R Rotman, Filling metric spaces, Duke Math. J. 171 (2022) 595 MR4382977
21 Y Liokumovich, F C Marques, A Neves, Weyl law for the volume spectrum, Ann. of Math. 187 (2018) 933 MR3779961
22 Y Liokumovich, B Staffa, Generic density of geodesic nets, Selecta Math. 30 (2024) 14 MR4688379
23 F C Marques, A Neves, Min-max theory and the Willmore conjecture, Ann. of Math. 179 (2014) 683 MR3152944
24 F C Marques, A Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math. 209 (2017) 577 MR3674223
25 F C Marques, A Neves, Morse index of multiplicity one min-max minimal hypersurfaces, Adv. Math. 378 (2021) 107527 MR4191255
26 F C Marques, A Neves, A Song, Equidistribution of minimal hypersurfaces for generic metrics, Invent. Math. 216 (2019) 421 MR3953507
27 L Mazurowski, A Weyl law for the p-Laplacian, J. Funct. Anal. 288 (2025) 110734
28 J T Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, 27, Princeton Univ. Press (1981) MR0626027
29 A Song, A dichotomy for minimal hypersurfaces in manifolds thick at infinity, Ann. Sci. École Norm. Sup. 56 (2023) 1085 MR4650156
30 A Song, Existence of infinitely many minimal hypersurfaces in closed manifolds, Ann. of Math. 197 (2023) 859 MR4564260
31 A Song, X Zhou, Generic scarring for minimal hypersurfaces along stable hypersurfaces, Geom. Funct. Anal. 31 (2021) 948 MR4317508
32 B Staffa, Bumpy metrics theorem for geodesic nets, J. Topol. Anal. (2023)
33 B Staffa, Parametric coarea inequality for 1-cycles, preprint (2024) arXiv:2410.23195
34 B Staffa, Weyl law for 1-cycles, preprint (2024) arXiv:2410.23192
35 S Wenger, A short proof of Gromov’s filling inequality, Proc. Amer. Math. Soc. 136 (2008) 2937 MR2399061
36 R E Williamson Jr., Cobordism of combinatorial manifolds, Ann. of Math. 83 (1966) 1 MR0184242
37 X Zhou, On the multiplicity one conjecture in min-max theory, Ann. of Math. 192 (2020) 767 MR4172621