Properly embedded minimal planar domains

Annals of Mathematics

Volumn 181, Issue 2, 2015, Pages 473-546

  Meeks, William H ^a   Pérez, Joaquín ^b   Ros, Antonio ^b  

^a UNIVERSITY OF MASSACHUSETTS   (United States)

^b UNIVERSITY OF GRANADA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84911380291     PISSN: 0003486X     EISSN: 19398980     Source Type: Journal    
DOI: 10.4007/annals.2015.181.2.2     Document Type: Article

References (57)

1. 3, Amer. J. Math. 98 (1976), 515-528. MR 0413172. Zbl 0332.53006. http://dx.doi.org/10.2307/2373899.
2. J. Bernstein and C. Breiner, Conformal structure of minimal surfaces with finite topology, Comment. Math. Helv. 86 (2011), 353-381. MR 2775132. Zbl 1213.53011. http://dx.doi.org/10.4171/CMH/226.
3. J. C. Borda, Eclaircissement sur les méthodes de trouver ler courbes qui jouissent de quelque propiété du maximum ou du minimum, Mém. Acad. Roy. Sci. Paris (1770), 551-565, presented in 1767.
4. M. Callahan, D. Hoffman, and W. H. Meeks, III, The structure of singlyperiodic minimal surfaces, Invent. Math. 99 (1990), 455-481. MR 1032877. Zbl 0695.53005. http://dx.doi.org/10.1007/BF01234428.
5. S. S. Chern and C. K. Peng, Lie groups and KdV equations, Manuscripta Math. 28 (1979), 207-217. MR 0535702. Zbl 0408.35074. http://dx.doi.org/10.1007/BF01647972.
6. T. H. Colding, C. De Lellis, and W. P. Minicozzi, II, Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications, Comm. Pure Appl. Math. 61 (2008), 1540-1602. MR 2444375. Zbl 05358518. http://dx.doi.org/10.1002/cpa.20232.
7. T. H. Colding and W. P. Minicozzi, II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. I. Estimates offthe axis for disks, Ann. of Math. 160 (2004), 27-68. MR 2119717. Zbl 1070.53031. http://dx.doi.org/10.4007/annals.2004.160.27.
8. 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. 160 (2004), 69-92. MR 2119718. Zbl 1070.53032. http://dx.doi.org/10.4007/annals.2004.160.69.
9. 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. 160 (2004), 523-572. MR 2123932. Zbl 1076.53068. http://dx.doi.org/10.4007/annals.2004.160.523.
10. 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. 160 (2004), 573-615. MR 2123933. Zbl 1076.53069. http://dx.doi.org/10.4007/annals.2004.160.573.
11. 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. 181 (2015), 1-153. http://dx.doi.org/10.4007/annals.2015.181.1.1.
12. 3, Ann. of Math. 145 (1997), 1-31. MR 1432035. Zbl 0886.53008. http://dx.doi.org/10.2307/2951822.
13. P. Collin, R. Kusner, W. H. Meeks, III, and H. Rosenberg, The topology, geometry and conformal structure of properly embedded minimal surfaces, J. Diérential Geom. 67 (2004), 377-393. MR 2153082. Zbl 1098.53006. Available at http://projecteuclid.org/euclid.jdg/1102536205.
14. A. Douady and R. Douady, Changements de cadres á partir des surfaces minimales, Cahier de DIDIREM 23 (1994), edited by IREM de Paris7.
15. N. Ejiri and M. Kotani, Index and at ends of minimal surfaces, Tokyo J. Math. 16 (1993), 37-48. MR 1223287. Zbl 0856.53013. http://dx.doi.org/10.3836/tjm/1270128981.
16. L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Propietate Gaudeates Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti, Harvard Univ. Press, Cambridge, MA, 1969, Opera omnia(1), 24, Fussli, Turici, 1952 (a source book in mathematics, partially translated by D. J. Struik); see pages 399-406.
17. Y. Fang, On minimal annuli in a slab, Comment. Math. Helv. 69 (1994), 417-430. MR 1289335. Zbl 0819.53006. http://dx.doi.org/10.1007/BF02564495.
18. C. Frohman and W. H. Meeks, III, The ordering theorem for the ends of properly embedded minimal surfaces, Topology 36 (1997), 605-617. MR 1422427. Zbl 0878.53008. http://dx.doi.org/10.1016/S0040-9383(96)00019-5.
19. F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies-an analytic approach, Bull. Amer. Math. Soc. 35 (1998), 271-317. MR 1638298. Zbl 0909.34073. http://dx.doi.org/10.1090/S0273-0979-98-00765-4.
20. D. Gilbarg and N. S. Trudinger, Elliptic Partial Diérential Equations of Second Order, second ed., Grundl. Math. Wissen. 224, Springer-Verlag, New York, 1983. MR 0737190. Zbl 0562.35001. 544
21. R. E. Goldstein and D. M. Petrich, The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), 3203-3206. MR 1135964. Zbl 0990.37519. http://dx.doi.org/10.1103/PhysRevLett.67.3203.
22. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure Appl. Math., Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 0507725. Zbl 0408.14001.
23. L. Hauswirth and F. Pacard, Higher genus Riemann minimal surfaces, Invent. Math. 169 (2007), 569-620. MR 2336041. Zbl 1129.53009. http://dx.doi.org/10.1007/s00222-007-0056-z.
24. D. Hoffman and W. H. Meeks, III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), 373-377. MR 1062966. Zbl 0722.53054. http://dx.doi.org/10.1007/BF01231506.
25. D. Hoffman, M. Traizet, and B. White, Helicoidal minimal surfaces of precribed genus II. arXiv 1304.6180.
26. D. Hoffman and B. White, Genus-one helicoids from a variational point of view, Comment. Math. Helv. 83 (2008), 767-813. MR 2442963. Zbl 1161.53009. http://dx.doi.org/10.4171/CMH/143.
27. N. Joshi, The second Painlevé hierarchy and the stationary KdV hierarchy, Publ. Res. Inst. Math. Sci. 40 (2004), 1039-1061. MR 2074710. Zbl 1063.33030. http://dx.doi.org/10.2977/prims/1145475502.
28. J. L. Lagrange, Essai d'une nouvelle méthode pour determiner les maxima et les minima des formules integrales indefinies, Miscellanea Taurinensia 2 325 (1760), 173-199.
29. R. B. Lockhart and R. C. McOwen, Elliptic diérential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985), 409-447. MR 0837256. Zbl 0615.58048. Available at http://www.numdam.org/item?id= ASNSP 1985 4 12 3 409 0.
30. F. J. López and A. Ros, On embedded complete minimal surfaces of genus zero, J. Diérential Geom. 33 (1991), 293-300. MR 1085145. Zbl 0719.53004. Available at http://projecteuclid.org/euclid.jdg/1214446040.
31. W. H. Meeks, III, The limit lamination metric for the Colding-Minicozzi minimal lamination, Illinois J. Math. 49 (2005), 645-658. MR 2164355. Zbl 1087. 53058. Available at http://projecteuclid.org/euclid.ijm/1258138037.
32. W. H. Meeks, III and J. Pérez, Conformal properties in classical minimal surface theory, in Surveys in Diérential Geometry. Vol. IX, Int. Press, Somerville, MA, 2004, pp. 275-335. MR 2195411. Zbl 1086.53007. http://dx.doi.org/10.4310/SDG.2004.v9.n1.a8.
33. W. H. Meeks, III and J. Pérez, Embedded minimal surfaces of finite topology. Available at http://wdb.ugr.es/~jperez/publications-by-joaquin-perez/.
34. W. H. Meeks, III, J. Pérez, and A. Ros, Uniqueness of the Riemann minimal examples, Invent. Math. 133 (1998), 107-132. MR 1626477. Zbl 0916.53004. http://dx.doi.org/10.1007/s002220050241.
35. W. H. Meeks, III, J. Pérez, and A. Ros, The geometry of minimal surfaces of finite genus. I. Curvature estimates and quasiperiodicity, J. Diérential Geom. 66 (2004), 1-45. MR 2128712. Zbl 1068.53012. Available at http://projecteuclid. org/euclid.jdg/1090415028.
36. 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. MR 2096796. Zbl 1070.53003. http://dx.doi.org/10.1007/s00222-004-0374-3.
37. W. H. Meeks, III, J. Pérez, and A. Ros, Bounds on the topology and index of classical minimal surfaces. Available at http://wdb.ugr.es/~jperez/publications-by-joaquin-perez/.
38. W. H. Meeks, III, J. Pérez, and A. Ros, The embedded Calabi-Yau conjectures for finite genus. Available at http://wdb.ugr.es/~jperez/publications-by-joaquin-perez/.
39. W. H. Meeks, III and H. Rosenberg, The maximum principle at infinity for minimal surfaces in at three manifolds, Comment. Math. Helv. 65 (1990), 255-270. MR 1057243. Zbl 0713.53008. http://dx.doi.org/10.1007/BF02566606.
40. W. H. Meeks, III and H. Rosenberg, The uniqueness of the helicoid, Ann. of Math. 161 (2005), 727-758. MR 2153399. Zbl 1102.53005. http://dx.doi.org/10.4007/annals.2005.161.727.
41. J. B. Meusnier, Mémoire sur la courbure des surfaces, Mém. Mathém. Phys. Acad. Sci. Paris, prés. par div. Savans 10 (1785), 477-510, presented in 1776.
42. S. Montiel and A. Ros, Schrödinger operators associated to a holomorphic map, in Global Diérential Geometry and Global Analysis (Berlin, 1990), Lecture Notes in Math. 1481, Springer-Verlag, New York, 1991, pp. 147-174. MR 1178529. Zbl 0744.58007. http://dx.doi.org/10.1007/BFb0083639.
43. R. Osserman, Global properties of minimal surfaces in E3 and En, Ann. of Math. 80 (1964), 340-364. MR 0179701. Zbl 0134.38502. http://dx.doi.org/10.2307/1970396.
44. R. Osserman, A Survey of Minimal Surfaces, second ed., Dover Publications, New York, 1986. MR 0852409. Zbl 0209.52901.
45. J. Pérez, On singly-periodic minimal surfaces with planar ends, Trans. Amer. Math. Soc. 349 (1997), 2371-2389. MR 1407709. Zbl 0882.53007. http://dx.doi. org/10.1090/S0002-9947-97-01911-9.
46. J. Pérez and A. Ros, The space of properly embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J. 45 (1996), 177-204. MR 1406689. Zbl 0864.53008. http://dx.doi.org/10.1512/iumj.1996.45.2053.
47. B. Riemann, Über die Fläche vom kleinsten Inhalt bei gegebener Begrenzung, Abh. Königl, d. Wiss. Göttingen, Mathem. Cl. 13 (1867), 3-52. JFM 01.0218.01.
48. B. Riemann, Ouevres Mathématiques de Riemann, Gauthiers-Villars, Paris, 1898.
49. R. Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, in Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111-126. MR 0795231. Zbl 0532.53042. 546
50. R. M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Diérential Geom. 18 (1983), 791-809 (1984). MR 0730928. Zbl 0575.53037. Available at http://projecteuclid.org/euclid.jdg/1214438183.
51. G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5-65. MR 0783348. Zbl 0592.35112. http://dx.doi.org/10.1007/BF02698802.
52. M. Shiffman, On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Ann. of Math. 63 (1956), 77-90. MR 0074695. Zbl 0070.16803. http://dx.doi.org/10.2307/1969991.
53. M. Traizet, An embedded minimal surface with no symmetries, J. Diérential Geom. 60 (2002), 103-153. MR 1924593. Zbl 1054.53014. Available at http://projecteuclid.org/euclid.jdg/1090351085.
54. M. Weber, D. Hoffman, and M. Wolf, An embedded genus-one helicoid, Ann. of Math. 169 (2009), 347-448. MR 2480608. Zbl 1213.49049. http://dx.doi.org/10.4007/annals.2009.169.347.
55. M. Weber and M. Wolf, Teichmüller theory and handle addition for minimal surfaces, Ann. of Math. 156 (2002), 713-795. MR 1954234. Zbl 1028.53009. http://dx.doi.org/10.2307/3597281.
56. R. Weikard, On rational and periodic solutions of stationary KdV equations, Doc. Math. 4 (1999), 109126. MR 1683290. Zbl 0972.35121. Available at http://www.math.uiuc.edu/documenta/vol-04/04.html.
57. B. White, The space of m-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J. 36 (1987), 567-602. MR 0905611. Zbl 0770.58005. http://dx.doi.org/10.1512/iumj.1987.36.36031.