The space of embedded minimal surfaces of fixed genus in a 3-manifold. V. Fixed genus.

*(English)*Zbl 1322.53059The following compactness questions naturally arise when considering a class of submanifolds of a fixed complete Riemannian manifold \((M, g)\):

(1) Given a sequence \(\Sigma_i\) of smoothly immersed submanifolds, to what extent can we say that, up to passing to a subsequence, the \(\Sigma_i\) converge to a limit \(\Sigma\)?

(2) Moreover, assuming that all the \(\Sigma_i\) have a property \(\mathcal{P}\), to what extent does this affect the answer to the previous question and to what extent does \(\Sigma\) have the property \(\mathcal{P}\)?

Answering this type of questions has important consequences because, often, when studying a class of submanifolds useful information can be obtained about the class by considering to what extent the class is compact and by analyzing how failure of compactness may happen.

The paper under review is the fifth of a series of papers containing Colding and Minicozzi’s recent work regarding compactness properties of embedded minimal surfaces in three-manifolds. They mainly study the failure of smooth convergence for sequences of embedded minimal surfaces in \(\mathbb{R}^3\) with bounded genus because, as they see, the key is to understand the structure of an embedded minimal planar domain in a ball in \(\mathbb{R}^3\), that is, the case when the surfaces have genus zero, since the general case of fixed genus requires only minor changes.

In the first four papers, they considered the case of minimal disks. Roughly speaking, they characterized the failure of smooth compactness for embedded minimal disks in \(\mathbb{R}^3\), with the failure being always modelled on the helicoid. Their proof utilized and combined many properties of embedded minimal surfaces in \(\mathbb{R}^3\) and, in particular, they proved the so-called one-sided curvature estimate, which gives a uniform curvature bound for an embedded minimal disk that is close to, and on one side of, a plane.

The focus in this paper is on non-simply connected planar domains. Sequences of planar domains which are not simply connected are, after passing to a subsequence, naturally divided into two separate cases depending on whether or not the topology is concentrating at points. These cases are distinguished as follows: A sequence of surfaces \(\Sigma_i^2\subset \mathbb{R}^3\) is called {uniformly locally simply connected} (or ULSC) if for each compact subset \(K\) of \(\mathbb{R}^3\), there exists a constant \(r_0 > 0\) (depending on \(K\)) so that for every \(x \in K\), all \(r \leq r_0\) and every surface \(\Sigma_i\), each connected component of \(B_{r}(x) \cap \Sigma_i\) is a disk. A point \(y\) in \(\mathbb{R}^3\) is said to be in \(\mathcal{S}_{\mathrm{ulsc}}\) if the curvature for the sequence \(\Sigma_i\) blows up at \(y\) and the sequence is ULSC in a neighborhood of \(y\). The authors first show that every sequence \(\Sigma_i\) has a subsequence that is either ULSC or for which \(\mathcal{S}_{\mathrm{ulsc}}\) is empty and then prove that these two different cases give two very different structures. Following their earlier papers on disks, they obtain two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The first of these says that any such surface without small necks can be obtained by gluing together two oppositely-oriented double spiral staircases, while the second theorem gives a pair of pants (a topological disk with two subdisks removed) decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves. Both of these structures occur as different extremes in the \(2\)-parameter family of minimal surfaces known as the Riemann examples.

(1) Given a sequence \(\Sigma_i\) of smoothly immersed submanifolds, to what extent can we say that, up to passing to a subsequence, the \(\Sigma_i\) converge to a limit \(\Sigma\)?

(2) Moreover, assuming that all the \(\Sigma_i\) have a property \(\mathcal{P}\), to what extent does this affect the answer to the previous question and to what extent does \(\Sigma\) have the property \(\mathcal{P}\)?

Answering this type of questions has important consequences because, often, when studying a class of submanifolds useful information can be obtained about the class by considering to what extent the class is compact and by analyzing how failure of compactness may happen.

The paper under review is the fifth of a series of papers containing Colding and Minicozzi’s recent work regarding compactness properties of embedded minimal surfaces in three-manifolds. They mainly study the failure of smooth convergence for sequences of embedded minimal surfaces in \(\mathbb{R}^3\) with bounded genus because, as they see, the key is to understand the structure of an embedded minimal planar domain in a ball in \(\mathbb{R}^3\), that is, the case when the surfaces have genus zero, since the general case of fixed genus requires only minor changes.

In the first four papers, they considered the case of minimal disks. Roughly speaking, they characterized the failure of smooth compactness for embedded minimal disks in \(\mathbb{R}^3\), with the failure being always modelled on the helicoid. Their proof utilized and combined many properties of embedded minimal surfaces in \(\mathbb{R}^3\) and, in particular, they proved the so-called one-sided curvature estimate, which gives a uniform curvature bound for an embedded minimal disk that is close to, and on one side of, a plane.

The focus in this paper is on non-simply connected planar domains. Sequences of planar domains which are not simply connected are, after passing to a subsequence, naturally divided into two separate cases depending on whether or not the topology is concentrating at points. These cases are distinguished as follows: A sequence of surfaces \(\Sigma_i^2\subset \mathbb{R}^3\) is called {uniformly locally simply connected} (or ULSC) if for each compact subset \(K\) of \(\mathbb{R}^3\), there exists a constant \(r_0 > 0\) (depending on \(K\)) so that for every \(x \in K\), all \(r \leq r_0\) and every surface \(\Sigma_i\), each connected component of \(B_{r}(x) \cap \Sigma_i\) is a disk. A point \(y\) in \(\mathbb{R}^3\) is said to be in \(\mathcal{S}_{\mathrm{ulsc}}\) if the curvature for the sequence \(\Sigma_i\) blows up at \(y\) and the sequence is ULSC in a neighborhood of \(y\). The authors first show that every sequence \(\Sigma_i\) has a subsequence that is either ULSC or for which \(\mathcal{S}_{\mathrm{ulsc}}\) is empty and then prove that these two different cases give two very different structures. Following their earlier papers on disks, they obtain two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The first of these says that any such surface without small necks can be obtained by gluing together two oppositely-oriented double spiral staircases, while the second theorem gives a pair of pants (a topological disk with two subdisks removed) decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves. Both of these structures occur as different extremes in the \(2\)-parameter family of minimal surfaces known as the Riemann examples.

Reviewer: Oscar J. Garay (Bilbao)

##### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

58D10 | Spaces of embeddings and immersions |

PDF
BibTeX
XML
Cite

\textit{T. H. Colding} and \textit{W. P. II. Minicozzi}, Ann. Math. (2) 181, No. 1, 1--153 (2015; Zbl 1322.53059)

**OpenURL**

##### References:

[1] | M. Berger, A Panoramic View of Riemannian Geometry, New York: Springer-Verlag, 2003. · Zbl 1038.53002 |

[2] | T. H. Colding and W. P. Minicozzi II, Minimal surfaces, New York: New York University, Courant Institute of Mathematical Sciences, 1999, vol. 4. · Zbl 0987.49025 |

[3] | T. H. Colding and W. P. Minicozzi II, ”Estimates for parametric elliptic integrands,” Int. Math. Res. Not., vol. 2002, p. no. 6, 291-297. · Zbl 1002.53035 |

[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., vol. 160, iss. 1, pp. 27-68, 2004. · Zbl 1070.53031 |

[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., vol. 160, iss. 1, pp. 69-92, 2004. · Zbl 1070.53032 |

[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., vol. 160, iss. 2, pp. 523-572, 2004. · Zbl 1076.53068 |

[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., vol. 160, iss. 2, pp. 573-615, 2004. · Zbl 1076.53069 |

[8] | T. H. Colding and W. P. Minicozzi II, ”Multi-valued minimal graphs and properness of disks,” Internat. Math. Res. Notes, vol. 2002, iss. 21, pp. 1111-1127. · Zbl 1008.58012 |

[9] | T. H. Colding and W. P. Minicozzi II, ”On the structure of embedded minimal annuli,” Int. Math. Res. Not., vol. 2002, iss. 29, p. no. 29, 1539-1552. · Zbl 1122.53300 |

[10] | T. H. Colding and W. P. Minicozzi II, ”The Calabi-Yau conjectures for embedded surfaces,” Ann. of Math., vol. 167, iss. 1, pp. 211-243, 2008. · Zbl 1142.53012 |

[11] | T. H. Colding and W. P. Minicozzi II, ”Embedded minimal disks,” in Global Theory of Minimal Surfaces, Providence, RI: Amer. Math. Soc., 2005, vol. 2, pp. 405-438. · Zbl 1109.53008 |

[12] | T. H. Colding and W. P. Minicozzi II, ”Disks that are double spiral staircases,” Notices Amer. Math. Soc., vol. 50, iss. 3, pp. 327-339, 2003. · Zbl 1100.53008 |

[13] | T. H. Colding and W. P. Minicozzi II, ”Embedded minimal disks: Proper versus nonproper-global versus local,” Trans. Amer. Math. Soc., vol. 356, iss. 1, pp. 283-289, 2004. · Zbl 1046.53001 |

[14] | T. H. Colding and W. P. Minicozzi II, ”Minimal annuli with and without slits,” J. Symplectic Geom., vol. 1, iss. 1, pp. 47-61, 2001. · Zbl 1087.53008 |

[15] | T. H. Colding and W. P. Minicozzi II, ”Complete properly embedded minimal surfaces in \(\mathbb R^3\),” Duke Math. J., vol. 107, iss. 2, pp. 421-426, 2001. · Zbl 1010.49025 |

[16] | T. H. Colding and W. P. Minicozzi II, ”Shapes of embedded minimal surfaces,” Proc. Natl. Acad. Sci. USA, vol. 103, iss. 30, pp. 11106-11111, 2006. · Zbl 1175.53008 |

[17] | M. do Carmo and C. K. Peng, ”Stable complete minimal surfaces in \({\mathbf R}^3\) are planes,” Bull. Amer. Math. Soc., vol. 1, iss. 6, pp. 903-906, 1979. · Zbl 0442.53013 |

[18] | F. Fiala, ”Le problème des isopérimètres sur les surfaces ouvertes à courbure positive,” Comment. Math. Helv., vol. 13, pp. 293-346, 1941. · Zbl 0025.23003 |

[19] | D. Fischer-Colbrie and R. Schoen, ”The structure of complete stable minimal surfaces in \(3\)-manifolds of nonnegative scalar curvature,” Comm. Pure Appl. Math., vol. 33, iss. 2, pp. 199-211, 1980. · Zbl 0439.53060 |

[20] | M. Gromov, ”Groups of polynomial growth and expanding maps,” Inst. Hautes Études Sci. Publ. Math., vol. 53, pp. 53-73, 1981. · Zbl 0474.20018 |

[21] | R. Gulliver and B. H. Lawson Jr., ”The structure of stable minimal hypersurfaces near a singularity,” in Geometric Measure Theory and the Calculus of Variations, Providence, RI: Amer. Math. Soc., 1986, vol. 44, pp. 213-237. · Zbl 0592.53005 |

[22] | R. Hardt and L. Simon, ”Boundary regularity and embedded solutions for the oriented Plateau problem,” Ann. of Math., vol. 110, iss. 3, pp. 439-486, 1979. · Zbl 0457.49029 |

[23] | P. Hartman, ”Geodesic parallel coordinates in the large,” Amer. J. Math., vol. 86, pp. 705-727, 1964. · Zbl 0128.16105 |

[24] | S. Hildebrandt, ”Boundary behavior of minimal surfaces,” Arch. Rational Mech. Anal., vol. 35, pp. 47-82, 1969. · Zbl 0183.39402 |

[25] | D. Hoffman, F. S. Wei, and H. Karcher, ”Adding handles to the helicoid,” Bull. Amer. Math. Soc., vol. 29, iss. 1, pp. 77-84, 1993. · Zbl 0787.53003 |

[26] | M. Weber, D. Hoffman, and M. Wolf, ”An embedded genus-one helicoid,” Ann. of Math., vol. 169, iss. 2, pp. 347-448, 2009. · Zbl 1213.49049 |

[27] | D. Hoffman and W. H. Meeks III, ”The strong halfspace theorem for minimal surfaces,” Invent. Math., vol. 101, iss. 2, pp. 373-377, 1990. · Zbl 0722.53054 |

[28] | D. Hoffman and B. White, ”Sequences of embedded minimal disks whose curvatures blow up on a prescribed subset of a line,” Comm. Anal. Geom., vol. 19, iss. 3, pp. 487-502, 2011. · Zbl 1244.53010 |

[29] | S. J. Kleene, ”A minimal lamination with Cantor set-like singularities,” Proc. Amer. Math. Soc., vol. 140, iss. 4, pp. 1423-1436, 2012. · Zbl 1239.53006 |

[30] | R. Langevin and H. Rosenberg, ”A maximum principle at infinity for minimal surfaces and applications,” Duke Math. J., vol. 57, iss. 3, pp. 819-828, 1988. · Zbl 0667.49024 |

[31] | W. H. Meeks III, ”Regularity of the singular set in the Colding-Minicozzi lamination theorem,” Duke Math. J., vol. 123, iss. 2, pp. 329-334, 2004. · Zbl 1086.53005 |

[32] | W. H. Meeks III, ”The limit lamination metric for the Colding-Minicozzi minimal lamination,” Illinois J. Math., vol. 49, iss. 2, pp. 645-658, 2005. · Zbl 1087.53058 |

[33] | W. H. Meeks III and J. Pérez, ”Conformal properties in classical minimal surface theory,” in Surveys in Differential Geometry. Vol. IX, Somerville, MA: Int. Press, 2004, vol. IX, pp. 275-335. · Zbl 1086.53007 |

[34] | 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., vol. 66, iss. 1, pp. 1-45, 2004. · Zbl 1068.53012 |

[35] | 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., vol. 158, iss. 2, pp. 323-341, 2004. · Zbl 1070.53003 |

[36] | W. H. Meeks III, J. Pérez, and A. Ros, The geometry of minimal surfaces of finite genus III; bounds on the topology and index of classical minimal surfaces. |

[37] | W. H. Meeks III, J. Pérez, and A. Ros, Properly embedded minimal planar domains. · Zbl 1315.53008 |

[38] | W. H. Meeks III, J. Pérez, and A. Ros, ”Limit leaves of an \(H\) lamination are stable,” J. Differential Geom., vol. 84, iss. 1, pp. 179-189, 2010. · Zbl 1197.53037 |

[39] | W. H. Meeks III, J. Pérez, and A. Ros, ”Liouville-type properties for embedded minimal surfaces,” Comm. Anal. Geom., vol. 14, iss. 4, pp. 703-723, 2006. · Zbl 1117.53009 |

[40] | W. H. Meeks III and H. Rosenberg, ”The uniqueness of the helicoid,” Ann. of Math., vol. 161, iss. 2, pp. 727-758, 2005. · Zbl 1102.53005 |

[41] | W. H. Meeks III and H. Rosenberg, ”The minimal lamination closure theorem,” Duke Math. J., vol. 133, iss. 3, pp. 467-497, 2006. · Zbl 1098.53007 |

[42] | W. H. Meeks III and M. Weber, ”Bending the helicoid,” Math. Ann., vol. 339, iss. 4, pp. 783-798, 2007. · Zbl 1156.53011 |

[43] | W. H. Meeks III and S. T. Yau, ”The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn’s lemma,” Topology, vol. 21, iss. 4, pp. 409-442, 1982. · Zbl 0489.57002 |

[44] | W. W. Meeks III and S. T. Yau, ”The existence of embedded minimal surfaces and the problem of uniqueness,” Math. Z., vol. 179, iss. 2, pp. 151-168, 1982. · Zbl 0479.49026 |

[45] | K. Nomizu and H. Ozeki, ”The existence of complete Riemannian metrics,” Proc. Amer. Math. Soc., vol. 12, pp. 889-891, 1961. · Zbl 0102.16401 |

[46] | J. Pérez, Limits by rescalings of minimal surfaces: Minimal laminations, curvature decay and local pictures. |

[47] | H. Rosenberg, ”Some recent developments in the theory of minimal surfaces in 3-manifolds,” in 24\({}^{o}\) Colóquio Brasileiro de Matemática. [24th Brazilian Mathematics Colloquium], Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA), 2003, p. iv. · Zbl 1064.53007 |

[48] | R. Schoen, ”Estimates for stable minimal surfaces in three-dimensional manifolds,” in Seminar on Minimal Submanifolds, Princeton, NJ: Princeton Univ. Press, 1983, vol. 103, pp. 111-126. · Zbl 0532.53042 |

[49] | R. M. Schoen, ”Uniqueness, symmetry, and embeddedness of minimal surfaces,” J. Differential Geom., vol. 18, iss. 4, pp. 791-809 (1984), 1983. · Zbl 0575.53037 |

[50] | K. Shiohama and M. Tanaka, ”An isoperimetric problem for infinitely connected complete open surfaces,” in Geometry of Manifolds, Boston: Academic Press, 1989, vol. 8, pp. 317-343. · Zbl 0697.53040 |

[51] | K. Shiohama and M. Tanaka, ”The length function of geodesic parallel circles,” in Progress in Differential Geometry, Tokyo: Math. Soc. Japan, 1993, vol. 22, pp. 299-308. · Zbl 0799.53052 |

[52] | B. Solomon, ”On foliations of \({\mathbf R}^{n+1}\) by minimal hypersurfaces,” Comment. Math. Helv., vol. 61, iss. 1, pp. 67-83, 1986. · Zbl 0601.53025 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.