Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 7, 3001–3510
Issue 6, 2483–2999
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
On the existence of minimal hypersurfaces with arbitrarily large area and Morse index

Yangyang Li

Geometry & Topology 26 (2022) 2713–2729
DOI: 10.2140/gt.2022.26.2713
Bibliography
1 F J Almgren Jr., The homotopy groups of the integral cycle groups, PhD thesis, Brown University (1962)
2 F J Almgren Jr., The theory of varifolds : a variational calculus in the large for the k–dimensional area integrand, lecture notes (1965)
3 L Ambrozio, A Carlotto, B Sharp, Compactness analysis for free boundary minimal hypersurfaces, Calc. Var. Partial Differential Equations 57 (2018) MR3740402
4 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
5 O Chodosh, C Mantoulidis, Minimal hypersurfaces with arbitrarily large area, Int. Math. Res. Not. 2021 (2021) 10841 MR4285736
6 T H Colding, N Hingston, Metrics without Morse index bounds, Duke Math. J. 119 (2003) 345 MR1997949
7 T H Colding, W P Minicozzi II, Examples of embedded minimal tori without area bounds, Int. Math. Res. Not. 1999 (1999) 1097 MR1728019
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 M A M Guaraco, Min-max for phase transitions and the existence of embedded minimal hypersurfaces, J. Differential Geom. 108 (2018) 91 MR3743704
10 J Hass, P Norbury, J H Rubinstein, Minimal spheres of arbitrarily high Morse index, Comm. Anal. Geom. 11 (2003) 425 MR2015753
11 K Irie, F C Marques, A Neves, Density of minimal hypersurfaces for generic metrics, Ann. of Math. 187 (2018) 963 MR3779962
12 M M C Li, X Zhou, Min-max theory for free boundary minimal hypersurfaces, I : Regularity theory, J. Differential Geom. 118 (2021) 487 MR4285846
13 Y Li, Existence of infinitely many minimal hypersurfaces in higher-dimensional closed manifolds with generic metrics, (2019) arXiv:1901.08440
14 Y Liokumovich, F C Marques, A Neves, Weyl law for the volume spectrum, Ann. of Math. 187 (2018) 933 MR3779961
15 F C Marques, A Neves, Min-max theory and the Willmore conjecture, Ann. of Math. 179 (2014) 683 MR3152944
16 F C Marques, A Neves, Morse index and multiplicity of min-max minimal hypersurfaces, Camb. J. Math. 4 (2016) 463 MR3572636
17 F C Marques, A Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math. 209 (2017) 577 MR3674223
18 F C Marques, A Neves, Morse index of multiplicity one min-max minimal hypersurfaces, Adv. Math. 378 (2021) MR4191255
19 F C Marques, A Neves, A Song, Equidistribution of minimal hypersurfaces for generic metrics, Invent. Math. 216 (2019) 421 MR3953507
20 J T Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, 27, Princeton Univ. Press (1981) MR626027
21 J T Pitts, J H Rubinstein, Applications of minimax to minimal surfaces and the topology of 3–manifolds, from: "Miniconference on geometry and partial differential equations, II" (editors J E Hutchinson, L M Simon), Proc. Centre Math. Anal. Austral. Nat. Univ. 12, The Australian National University (1987) 137 MR924434
22 R Schoen, L Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981) 741 MR634285
23 B Sharp, Compactness of minimal hypersurfaces with bounded index, J. Differential Geom. 106 (2017) 317 MR3662994
24 A Song, Existence of infinitely many minimal hypersurfaces in closed manifolds, preprint (2018) arXiv:1806.08816
25 B White, The maximum principle for minimal varieties of arbitrary codimension, Comm. Anal. Geom. 18 (2010) 421 MR2747434
26 B White, On the bumpy metrics theorem for minimal submanifolds, Amer. J. Math. 139 (2017) 1149 MR3689325
27 S T Yau, Problem section, from: "Seminar on differential geometry" (editor S T Yau), Ann. of Math. Stud. 102, Princeton Univ. Press (1982) 669 MR645762
28 X Zhou, On the multiplicity one conjecture in min-max theory, Ann. of Math. 192 (2020) 767 MR4172621
29 X Zhou, J Zhu, Existence of hypersurfaces with prescribed mean curvature, I : Generic min-max, Camb. J. Math. 8 (2020) 311 MR4091027