Index growth not imputable to topology

Carlotto, Alessandro; Schulz, Mario; Wiygul, David

doi:10.1090/proc/17151

Index growth not imputable to topology
HTML articles powered by AMS MathViewer

by Alessandro Carlotto, Mario B. Schulz and David Wiygul;

Proc. Amer. Math. Soc. 153 (2025), 1787-1801

DOI: https://doi.org/10.1090/proc/17151

Published electronically: February 10, 2025

HTML | PDF | Request permission

Abstract:

We employ partitioning methods, in the spirit of Montiel–Ros but here recast for general actions of compact Lie groups, to prove effective lower bounds on the Morse index of certain families of closed minimal hypersurfaces in the round four-dimensional sphere, and of free boundary minimal hypersurfaces in the Euclidean four-dimensional ball. Our analysis reveals, in particular, phenomena of linear index growth for sequences of minimal hypersurfaces of fixed topological type, in strong contrast to the three-dimensional scenario.

References

Hilário Alencar, Minimal hypersurfaces of $\textbf {R}^{2m}$ invariant by $\textrm {SO}(m)\times \textrm {SO}(m)$, Trans. Amer. Math. Soc. 337 (1993), no. 1, 129–141. MR 1091229, DOI 10.1090/S0002-9947-1993-1091229-1
Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp, Compactness analysis for free boundary minimal hypersurfaces, Calc. Var. Partial Differential Equations 57 (2018), no. 1, Paper No. 22, 39. MR 3740402, DOI 10.1007/s00526-017-1281-y
Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp, Comparing the Morse index and the first Betti number of minimal hypersurfaces, J. Differential Geom. 108 (2018), no. 3, 379–410. MR 3770846, DOI 10.4310/jdg/1519959621
Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp, Index estimates for free boundary minimal hypersurfaces, Math. Ann. 370 (2018), no. 3-4, 1063–1078. MR 3770163, DOI 10.1007/s00208-017-1549-8
Alessandro Carlotto and Mario B. Schulz, Minimal hypertori in the four-dimensional sphere, Ars Inven. Anal. (2023), Paper No. 8, 33. MR 4645955
A. Carlotto, M. B. Schulz, and D. Wiygul, Spectral estimates for free boundary minimal surfaces via Montiel–Ros partitioning methods, Anal. PDE (to appear), arXiv:2301.03055, 2023.
Elie Cartan, Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45 (1939), 335–367 (French). MR 169, DOI 10.1007/BF01580289
Jaigyoung Choe, Index, vision number and stability of complete minimal surfaces, Arch. Rational Mech. Anal. 109 (1990), no. 3, 195–212. MR 1025170, DOI 10.1007/BF00375088
Hyeong In Choi and Ai Nung Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom. 18 (1983), no. 3, 559–562. MR 723817
Santiago Cordero-Misteli and Giada Franz, Estimating the Morse index of free boundary minimal hypersurfaces through covering arguments, J. Reine Angew. Math. 807 (2024), 187–220. MR 4698495, DOI 10.1515/crelle-2023-0087
Norio Ejiri and Mario Micallef, Comparison between second variation of area and second variation of energy of a minimal surface, Adv. Calc. Var. 1 (2008), no. 3, 223–239. MR 2458236, DOI 10.1515/ACV.2008.009
Brian Freidin, Mamikon Gulian, and Peter McGrath, Free boundary minimal surfaces in the unit ball with low cohomogeneity, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1671–1683. MR 3601558, DOI 10.1090/proc/13424
Alexander Grigor′yan, Yuri Netrusov, and Shing-Tung Yau, Eigenvalues of elliptic operators and geometric applications, Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., vol. 9, Int. Press, Somerville, MA, 2004, pp. 147–217. MR 2195408, DOI 10.4310/SDG.2004.v9.n1.a5
Wu-Yi Hsiang, Minimal cones and the spherical Bernstein problem. I, Ann. of Math. (2) 118 (1983), no. 1, 61–73. MR 707161, DOI 10.2307/2006954
Wu-yi Hsiang and H. Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1–38. MR 298593
Nikolaos Kapouleas and David Wiygul, The index and nullity of the Lawson surfaces $\xi _{g,1}$, Camb. J. Math. 8 (2020), no. 2, 363–405. MR 4091028, DOI 10.4310/CJM.2020.v8.n2.a3
Vanderson Lima, Bounds for the Morse index of free boundary minimal surfaces, Asian J. Math. 26 (2022), no. 2, 227–252. MR 4557080, DOI 10.4310/ajm.2022.v26.n2.a3
Yevgeny Liokumovich, Fernando C. Marques, and André Neves, Weyl law for the volume spectrum, Ann. of Math. (2) 187 (2018), no. 3, 933–961. MR 3779961, DOI 10.4007/annals.2018.187.3.7
Sebastián Montiel and Antonio Ros, Schrödinger operators associated to a holomorphic map, Global differential geometry and global analysis (Berlin, 1990) Lecture Notes in Math., vol. 1481, Springer, Berlin, 1991, pp. 147–174. MR 1178529, DOI 10.1007/BFb0083639
André Neves, New applications of min-max theory, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 939–957. MR 3728646
Oscar Perdomo, Low index minimal hypersurfaces of spheres, Asian J. Math. 5 (2001), no. 4, 741–749. MR 1913819, DOI 10.4310/AJM.2001.v5.n4.a8
Pam Sargent, Index bounds for free boundary minimal surfaces of convex bodies, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2467–2480. MR 3626504, DOI 10.1090/proc/13442
Alessandro Savo, Index bounds for minimal hypersurfaces of the sphere, Indiana Univ. Math. J. 59 (2010), no. 3, 823–837. MR 2779062, DOI 10.1512/iumj.2010.59.4013
Ben Sharp, Compactness of minimal hypersurfaces with bounded index, J. Differential Geom. 106 (2017), no. 2, 317–339. MR 3662994, DOI 10.4310/jdg/1497405628
Graham Smith, Ari Stern, Hung Tran, and Detang Zhou, On the Morse index of higher-dimensional free boundary minimal catenoids, Calc. Var. Partial Differential Equations 60 (2021), no. 6, Paper No. 208, 44. MR 4305430, DOI 10.1007/s00526-021-02049-8
Bruce Solomon, The harmonic analysis of cubic isoparametric minimal hypersurfaces. I. Dimensions $3$ and $6$, Amer. J. Math. 112 (1990), no. 2, 157–203. MR 1047297, DOI 10.2307/2374713
Antoine Song, Morse index, Betti numbers, and singular set of bounded area minimal hypersurfaces, Duke Math. J. 172 (2023), no. 11, 2073–2147. MR 4627248, DOI 10.1215/00127094-2023-0012
Francisco Urbano, Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc. 108 (1990), no. 4, 989–992. MR 1007516, DOI 10.1090/S0002-9939-1990-1007516-1
Zhichao Wang, Existence of infinitely many free boundary minimal hypersurfaces, J. Differential Geom. 126 (2024), no. 1, 363–399. MR 4704552, DOI 10.4310/jdg/1707767341
Xin Zhou, On the multiplicity one conjecture in min-max theory, Ann. of Math. (2) 192 (2020), no. 3, 767–820. MR 4172621, DOI 10.4007/annals.2020.192.3.3