The systole of large genus minimal surfaces in positive Ricci curvature

Matthiesen, Henrik¹ ; Siffert, Anna²

¹Max Planck Institute for Mathematics, Bonn, Germany, Department of Mathematics, University of Chicago, Chicago, IL, United States
²Mathematisches Institut, University of Münster, Münster, Germany

Geometry & topology, Tome 29 (2025) no. 4, pp. 1819-1849

Résumé

We use Colding–Minicozzi lamination theory to show that the systole, and more generally any homology systole, of a sequence of embedded minimal surfaces in an ambient three-manifold of positive Ricci curvature tends to zero as the genus becomes unbounded.

Détail

DOI : 10.2140/gt.2025.29.1819

Keywords: systole, minimal surfaces

Bibliographie

[1] N S Aiex, Non-compactness of the space of minimal hypersurfaces, Math. Ann. 370 (2018) 191 | DOI

[2] F Balacheff, H Parlier, S Sabourau, Short loop decompositions of surfaces and the geometry of Jacobians, Geom. Funct. Anal. 22 (2012) 37 | DOI

[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, D Ketover, D Maximo, Minimal hypersurfaces with bounded index, Invent. Math. 209 (2017) 617 | DOI

[5] H I Choi, R Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985) 387 | DOI

[6] H I Choi, A N Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom. 18 (1983) 559

[7] T H Colding, W P Minicozzi Ii, Embedded minimal disks: proper versus nonproper, global versus local, Trans. Amer. Math. Soc. 356 (2004) 283 | DOI

[8] T H Colding, W P Minicozzi Ii, The space of embedded minimal surfaces of fixed genus in a 3-manifold, I : Estimates off the axis for disks, Ann. of Math. 160 (2004) 27 | DOI

[9] T H Colding, W P Minicozzi Ii, The space of embedded minimal surfaces of fixed genus in a 3-manifold, II : Multi-valued graphs in disks, Ann. of Math. 160 (2004) 69 | DOI

[10] T H Colding, W P Minicozzi Ii, The space of embedded minimal surfaces of fixed genus in a 3-manifold, III : Planar domains, Ann. of Math. 160 (2004) 523 | DOI

[11] T H Colding, W P Minicozzi Ii, The space of embedded minimal surfaces of fixed genus in a 3-manifold, IV : Locally simply connected, Ann. of Math. 160 (2004) 573 | DOI

[12] T H Colding, W P Minicozzi Ii, The Calabi–Yau conjectures for embedded surfaces, Ann. of Math. 167 (2008) 211 | DOI

[13] T H Colding, W P Minicozzi Ii, The space of embedded minimal surfaces of fixed genus in a 3-manifold, V : Fixed genus, Ann. of Math. 181 (2015) 1 | DOI

[14] D Fischer-Colbrie, R Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980) 199 | DOI

[15] M Freedman, J Hass, P Scott, Least area incompressible surfaces in 3-manifolds, Invent. Math. 71 (1983) 609 | DOI

[16] M Gromov, Systoles and intersystolic inequalities, from: "Actes de la Table Ronde de Géométrie Différentielle" (editor A L Besse), Sémin. Congr. 1, Soc. Math. France (1996) 291

[17] R Hardt, L Simon, Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. of Math. 110 (1979) 439 | DOI

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

[19] H B Lawson Jr., Complete minimal surfaces in S3, Ann. of Math. 92 (1970) 335 | DOI

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

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

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

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

[24] W H Meeks Iii, J Pérez, A Ros, Limit leaves of an H lamination are stable, J. Differential Geom. 84 (2010) 179

[25] D A Neumann, 3-manifolds fibering over S1, Proc. Amer. Math. Soc. 58 (1976) 353 | DOI

[26] R Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, from: "Seminar on minimal submanifolds" (editor E Bombieri), Ann. of Math. Stud. 103, Princeton Univ. Press (1983) 111

[27] R Schoen, S T Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. 110 (1979) 127 | DOI

[28] R Schoen, S T Yau, The existence of a black hole due to condensation of matter, Comm. Math. Phys. 90 (1983) 575 | DOI

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

[30] J L Tollefson, 3-manifolds fibering over S1 with nonunique connected fiber, Proc. Amer. Math. Soc. 21 (1969) 79 | DOI

[31] B White, Curvatures of embedded minimal disks blow up on subsets of C1 curves, J. Differential Geom. 100 (2015) 389

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