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Structure theorems for singular minimal laminations

  • William H. Meeks III EMAIL logo , Joaquín Pérez ORCID logo and Antonio Ros
Published/Copyright: January 20, 2019

Abstract

We apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly singular minimal laminations of 3. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in 3, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in 3 with finite genus and a countable number of ends is proper.

Award Identifier / Grant number: DMS-1309236

Funding statement: William H. Meeks III was supported by the NSF under Award No. DMS-1309236. The second and third authors were partially supported by MINECO/FEDER grants no. MTM2014-52368-P and MTM2017-89677-P.

References

[1] W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. 10.2307/1970868Search in Google Scholar

[2] T. H. Colding and W. P. Minicozzi II, Multivalued minimal graphs and properness of disks, Int. Math. Res. Not. IMRN 2002 (2002), no. 21, 1111–1127. 10.1155/S107379280211107XSearch in Google Scholar

[3] T. H. Colding and W. P. Minicozzi II, Embedded minimal disks: Proper versus nonproper—global versus local, Trans. Amer. Math. Soc. 356 (2004), no. 1, 283–289. 10.1090/S0002-9947-03-03230-6Search in Google Scholar

[4] T. H. Colding and 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. (2) 160 (2004), 27–68. 10.4007/annals.2004.160.27Search in Google Scholar

[5] T. H. Colding and 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. (2) 160 (2004), 69–92. 10.4007/annals.2004.160.69Search in Google Scholar

[6] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. III: Planar domains, Ann. of Math. (2) 160 (2004), 523–572. 10.4007/annals.2004.160.523Search in Google Scholar

[7] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV: Locally simply-connected, Ann. of Math. (2) 160 (2004), 573–615. 10.4007/annals.2004.160.573Search in Google Scholar

[8] T. H. Colding and W. P. Minicozzi II, The Calabi–Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211–243. 10.4007/annals.2008.167.211Search in Google Scholar

[9] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. V: Fixed genus, Ann. of Math. (2) 181 (2015), no. 1, 1–153. 10.4007/annals.2015.181.1.1Search in Google Scholar

[10] P. Collin, Topologie et courbure des surfaces minimales proprement plongées de 𝐑3, Ann. of Math. (2) 145 (1997), no. 1, 1–31. 10.2307/2951822Search in Google Scholar

[11] P. Collin, R. Kusner, W. H. Meeks III and H. Rosenberg, The topology, geometry and conformal structure of properly embedded minimal surfaces, J. Differential Geom. 67 (2004), no. 2, 377–393. 10.4310/jdg/1102536205Search in Google Scholar

[12] P. E. Ehrlich, Continuity properties of the injectivity radius function, Compos. Math. 29 (1974), 151–178. Search in Google Scholar

[13] A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N. S.) 36 (1999), no. 2, 135–249. 10.1090/S0273-0979-99-00776-4Search in Google Scholar

[14] M. Grüter, Regularity of weak H-surfaces, J. reine angew. Math. 329 (1981), 1–15. 10.1515/crll.1981.329.1Search in Google Scholar

[15] R. Harvey and B. Lawson, Extending minimal varieties, Invent. Math. 28 (1975), 209–226. 10.1007/BF01425557Search in Google Scholar

[16] D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373–377. 10.1007/BF01231506Search in Google Scholar

[17] F. J. López and A. Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), no. 1, 293–300. 10.4310/jdg/1214446040Search in Google Scholar

[18] W. H. Meeks III, Regularity of the singular set in the Colding–Minicozzi lamination theorem, Duke Math. J. 123 (2004), no. 2, 329–334. 10.1215/S0012-7094-04-12324-3Search in Google Scholar

[19] W. H. Meeks III, The limit lamination metric for the Colding–Minicozzi minimal lamination, Illinois J. Math. 49 (2005), no. 2, 645–658. 10.1215/ijm/1258138037Search in Google Scholar

[20] W. H. Meeks III and J. Pérez, The classical theory of minimal surfaces, Bull. Amer. Math. Soc. (N. S.) 48 (2011), no. 3, 325–407. 10.1090/S0273-0979-2011-01334-9Search in Google Scholar

[21] W. H. Meeks III, J. Pérez and A. Ros, The geometry of minimal surfaces of finite genus. I: Curvature estimates and quasiperiodicity, J. Differential Geom. 66 (2004), 1–45. 10.4310/jdg/1090415028Search in Google Scholar

[22] W. H. Meeks III, J. Pérez and A. Ros, The geometry of minimal surfaces of finite genus II: Nonexistence of one limit end examples, Invent. Math. 158 (2004), 323–341. 10.1007/s00222-004-0374-3Search in Google Scholar

[23] W. H. Meeks III, J. Pérez and A. Ros, Stable constant mean curvature surfaces, Handbook of geometrical analysis. Vol. 1, International Press, Somerville (2008), 301–380. Search in Google Scholar

[24] W. H. Meeks III, J. Pérez and A. Ros, Limit leaves of an H lamination are stable, J. Differential Geom. 84 (2010), no. 1, 179–189. 10.4310/jdg/1271271797Search in Google Scholar

[25] W. H. Meeks III, J. Pérez and A. Ros, Properly embedded minimal planar domains, Ann. of Math. (2) 181 (2015), no. 2, 473–546. 10.4007/annals.2015.181.2.2Search in Google Scholar

[26] W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint (2016), https://arxiv.org/abs/1605.02501. Search in Google Scholar

[27] W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362. 10.4310/jdg/1463404121Search in Google Scholar

[28] W. H. Meeks III, J. Pérez and A. Ros, The classification of CMC foliations of 3 and 𝕊3 with countably many singularities, Amer. J. Math. 138 (2016), no. 5, 1347–1382. 10.1353/ajm.2016.0040Search in Google Scholar

[29] W. H. Meeks III, J. Pérez and A. Ros, The dynamics theorem for properly embedded minimal surfaces, Math. Ann. 365 (2016), no. 3–4, 1069–1089. 10.1007/s00208-015-1311-zSearch in Google Scholar

[30] W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint (2018), https://arxiv.org/abs/1806.03104. 10.1215/00127094-2020-0087Search in Google Scholar

[31] W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565. 10.4310/jdg/1531188195Search in Google Scholar

[32] W. H. Meeks III and H. Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), no. 2, 727–758. 10.4007/annals.2005.161.727Search in Google Scholar

[33] W. H. Meeks III and H. Rosenberg, The minimal lamination closure theorem, Duke Math. J. 133 (2006), no. 3, 467–497. 10.1215/S0012-7094-06-13332-XSearch in Google Scholar

[34] W.  Meeks III, L. Simon and S. T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. 10.2307/2007026Search in Google Scholar

[35] W. W. Meeks III and S. T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), no. 2, 151–168. 10.1007/BF01214308Search in Google Scholar

[36] T. Sakai, On continuity of injectivity radius function, Math. J. Okayama Univ. 25 (1983), no. 1, 91–97. Search in Google Scholar

Received: 2018-04-24
Revised: 2018-10-24
Published Online: 2019-01-20
Published in Print: 2020-06-01

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