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

Abstract

This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for \itnonsimply connected embedded minimal surfaces of any given fixed genus.
The first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely-oriented double spiral staircases.
The second gives a pair of pants decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves. After the cutting, we are left with graphical pieces that are defined over a disk with either one or two sub-disks removed (a topological disk with two sub-disks removed is called a pair of pants).
Both of these structures occur as different extremes in the two-parameter family of minimal surfaces known as the Riemann examples.

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  • [B] Go to document M. Berger, A Panoramic View of Riemannian Geometry, New York: Springer-Verlag, 2003.
    @book{B, mrkey = {2002701},
      author = {Berger, Marcel},
      title = {A Panoramic View of {R}iemannian Geometry},
      publisher = {Springer-Verlag},
      year = {2003},
      pages = {xxiv+824},
      isbn = {3-540-65317-1},
      mrclass = {53-02 (53C20 53C21 53C55 58J20)},
      mrnumber = {2002701},
      mrreviewer = {J{ü}rgen Eichhorn},
      doi = {10.1007/978-3-642-18245-7},
      address = {New York},
      zblnumber = {1038.53002},
      }
  • [CM1] T. H. Colding and W. P. Minicozzi II, Minimal surfaces, New York: New York University, Courant Institute of Mathematical Sciences, 1999, vol. 4.
    @book{CM1, mrkey = {1683966},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Minimal surfaces},
      series = {Courant Lecture Notes in Math.},
      volume = {4},
      publisher = {New York University, Courant Institute of Mathematical Sciences},
      ADDRESS={New York},
      year = {1999},
      pages = {viii+124},
      ISBN = {0-9658703-3-2},
      mrclass = {49Q05 (49-02 53A10 58E12)},
      mrnumber = {1683966},
      zblnumber = {0987.49025},
      }
  • [CM2] Go to document T. H. Colding and W. P. Minicozzi II, "Estimates for parametric elliptic integrands," Int. Math. Res. Not., vol. 2002, p. no. 6, 291-297.
    @article{CM2, key = {CM02a},
      mrkey = {1877004},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Estimates for parametric elliptic integrands},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      pages = {no.~6, 291--297},
      issn = {1073-7928},
      mrclass = {53C21 (58E12)},
      mrnumber = {1877004},
      mrreviewer = {Man Chun Leung},
      doi = {10.1155/S1073792802106106},
      volume = {2002},
      sortyear = {2002},
      zblnumber = {1002.53035},
      }
  • [CM3] Go to document 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.
    @article{CM3, mrkey = {2119717},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {I}. {E}stimates off the axis for disks},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {160},
      year = {2004},
      number = {1},
      pages = {27--68},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10},
      mrnumber = {2119717},
      mrreviewer = {Oliver C. Schn{ü}rer},
      zblnumber = {1070.53031},
      doi = {10.4007/annals.2004.160.27},
      }
  • [CM4] Go to document 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.
    @article{CM4, mrkey = {2119718},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {II}. {M}ulti-valued graphs in disks},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {160},
      year = {2004},
      number = {1},
      pages = {69--92},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10},
      mrnumber = {2119718},
      mrreviewer = {Oliver C. Schn{ü}rer},
      zblnumber = {1070.53032},
      doi = {10.4007/annals.2004.160.69},
      }
  • [CM5] Go to document 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.
    @article{CM5, mrkey = {2123932},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {III}. {P}lanar domains},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {160},
      year = {2004},
      number = {2},
      pages = {523--572},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10 (53C42)},
      mrnumber = {2123932},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1076.53068},
      doi = {10.4007/annals.2004.160.523},
      }
  • [CM6] Go to document 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.
    @article{CM6, mrkey = {2123933},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {IV}. {L}ocally simply connected},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {160},
      year = {2004},
      number = {2},
      pages = {573--615},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10 (53C42)},
      mrnumber = {2123933},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1076.53069},
      doi = {10.4007/annals.2004.160.573},
      }
  • [CM7] Go to document 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.
    @article{CM7, key = {CM02b},
      mrkey = {1904463},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Multi-valued minimal graphs and properness of disks},
      journal = {Internat. Math. Res. Notes},
      fjournal = {International Mathematics Research Notices},
      volume = {2002},
      sortyear = {2002},
      number = {21},
      pages = {1111--1127},
      MRCLASS = {53A10 (35J60 49Q05 58E12)},
      mrnumber = {1904463},
      zblnumber = {1008.58012},
      doi = {10.1155/S107379280211107X},
      }
  • [CM8] Go to document 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.
    @article{CM8, mrkey = {1907204},
      key = {CM02c},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {On the structure of embedded minimal annuli},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      sortyear = {2002},
      number = {29},
      pages = {no.~29, 1539--1552},
      issn = {1073-7928},
      mrclass = {53A10},
      mrnumber = {1907204},
      mrreviewer = {Man Chun Leung},
      doi = {10.1155/S1073792802112128},
      volume = {2002},
      zblnumber = {1122.53300},
      }
  • [CM10] Go to document 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.
    @article{CM10, mrkey = {2373154},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {The {C}alabi-{Y}au conjectures for embedded surfaces},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {167},
      year = {2008},
      number = {1},
      pages = {211--243},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10 (53C42)},
      mrnumber = {2373154},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1142.53012},
      doi = {10.4007/annals.2008.167.211},
      }
  • [CM13] 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.
    @incollection{CM13, mrkey = {2167270},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Embedded minimal disks},
      booktitle = {Global Theory of Minimal Surfaces},
      series = {Clay Math. Proc.},
      volume = {2},
      pages = {405--438},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2005},
      mrclass = {53A10 (53C42)},
      mrnumber = {2167270},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1109.53008},
      }
  • [CM14] 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.
    @article{CM14, mrkey = {1954009},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Disks that are double spiral staircases},
      journal = {Notices Amer. Math. Soc.},
      fjournal = {Notices of the American Mathematical Society},
      volume = {50},
      year = {2003},
      number = {3},
      pages = {327--339},
      issn = {0002-9920},
      coden = {AMNOAN},
      mrclass = {53A10 (49Q05)},
      mrnumber = {1954009},
      mrreviewer = {J. E. Brothers},
      zblnumber = {1100.53008},
      }
  • [CM15] Go to document 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.
    @article{CM15, mrkey = {2020033},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Embedded minimal disks: Proper versus nonproper---global versus local},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {356},
      year = {2004},
      number = {1},
      pages = {283--289},
      issn = {0002-9947},
      coden = {TAMTAM},
      mrclass = {53A10 (49Q05)},
      mrnumber = {2020033},
      mrreviewer = {Michael John Dorff},
      zblnumber = {1046.53001},
      doi = {10.1090/S0002-9947-03-03230-6},
      }
  • [CM9] Go to document 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.
    @article{CM9, mrkey = {1959578},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Minimal annuli with and without slits},
      journal = {J. Symplectic Geom.},
      fjournal = {The Journal of Symplectic Geometry},
      volume = {1},
      year = {2001},
      number = {1},
      pages = {47--61},
      issn = {1527-5256},
      mrclass = {53A10},
      mrnumber = {1959578},
      mrreviewer = {Zhongmin Shen},
      url = {http://projecteuclid.org/euclid.jsg/1092316298},
      zblnumber = {1087.53008},
      }
  • [CM16] Go to document T. H. Colding and W. P. Minicozzi II, "Complete properly embedded minimal surfaces in $\Bbb R^3$," Duke Math. J., vol. 107, iss. 2, pp. 421-426, 2001.
    @article{CM16, mrkey = {1823052},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Complete properly embedded minimal surfaces in {$\bold R\sp 3$}},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {107},
      year = {2001},
      number = {2},
      pages = {421--426},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {53A10 (53C21)},
      mrnumber = {1823052},
      mrreviewer = {Man Chun Leung},
      doi = {10.1215/S0012-7094-01-10726-6},
      zblnumber = {1010.49025},
      }
  • [CM17] Go to document 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.
    @article{CM17, mrkey = {2242650},
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Shapes of embedded minimal surfaces},
      journal = {Proc. Natl. Acad. Sci. USA},
      fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
      volume = {103},
      year = {2006},
      number = {30},
      pages = {11106--11111},
      issn = {1091-6490},
      coden = {PNASFB},
      mrclass = {53A10},
      mrnumber = {2242650},
      mrreviewer = {Magdalena Daniela Toda},
      doi = {10.1073/pnas.0510379103},
      zblnumber = {1175.53008},
      }
  • [DoPe] Go to document M. do Carmo and C. K. Peng, "Stable complete minimal surfaces in ${\bf R}^{3}$ are planes," Bull. Amer. Math. Soc., vol. 1, iss. 6, pp. 903-906, 1979.
    @article{DoPe, mrkey = {0546314},
      key = {dCP79},
      author = {{do Carmo},
      M. and Peng, C. K.},
      title = {Stable complete minimal surfaces in {${\bf R}\sp{3}$} are planes},
      journal = {Bull. Amer. Math. Soc.},
      fjournal = {American Mathematical Society. Bulletin. New Series},
      volume = {1},
      year = {1979},
      number = {6},
      pages = {903--906},
      issn = {0273-0979},
      coden = {BAMOAD},
      mrclass = {53A10 (53C42)},
      mrnumber = {0546314},
      mrreviewer = {F. J. Almgren, Jr.},
      zblnumber = {0442.53013},
      doi = {10.1090/S0273-0979-1979-14689-5},
      }
  • [Fa] Go to document F. Fiala, "Le problème des isopérimètres sur les surfaces ouvertes à courbure positive," Comment. Math. Helv., vol. 13, pp. 293-346, 1941.
    @article{Fa, mrkey = {0006422},
      author = {Fiala, F.},
      title = {Le problème des isopérimètres sur les surfaces ouvertes à courbure positive},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {13},
      year = {1941},
      pages = {293--346},
      issn = {0010-2571},
      mrclass = {52.0X},
      mrnumber = {0006422},
      mrreviewer = {J. J. Stoker},
      doi = {10.1007/BF01378068},
      zblnumber = {0025.23003},
      }
  • [FiSc] Go to document 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.
    @article{FiSc, mrkey = {0562550},
      author = {Fischer-Colbrie, Doris and Schoen, Richard},
      title = {The structure of complete stable minimal surfaces in {$3$}-manifolds of nonnegative scalar curvature},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {33},
      year = {1980},
      number = {2},
      pages = {199--211},
      issn = {0010-3640},
      coden = {CPAMAT},
      mrclass = {53C40 (58E12)},
      mrnumber = {0562550},
      mrreviewer = {Themistocles M. Rassias},
      zblnumber = {0439.53060},
      doi = {10.1002/cpa.3160330206},
      }
  • [Gr] Go to document M. Gromov, "Groups of polynomial growth and expanding maps," Inst. Hautes Études Sci. Publ. Math., vol. 53, pp. 53-73, 1981.
    @article{Gr, mrkey = {0623534},
      author = {Gromov, Mikhael},
      title = {Groups of polynomial growth and expanding maps},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      volume = {53},
      year = {1981},
      pages = {53--73},
      issn = {0073-8301},
      coden = {PMIHA6},
      mrclass = {53C20 (22E40 58F15)},
      mrnumber = {0623534},
      mrreviewer = {J. A. Wolf},
      zblnumber = {0474.20018},
      DOI = {10.1007/BF02698687},
      }
  • [GuLa] Go to document 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.
    @incollection{GuLa, mrkey = {0840275},
      author = {Gulliver, Robert and Lawson, Jr., H. Blaine},
      title = {The structure of stable minimal hypersurfaces near a singularity},
      booktitle = {Geometric Measure Theory and the Calculus of Variations},
      venue = {{A}rcata, {C}alif., 1984},
      series = {Proc. Sympos. Pure Math.},
      volume = {44},
      pages = {213--237},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1986},
      mrclass = {53C42 (53A10 58G30)},
      mrnumber = {0840275},
      mrreviewer = {Michael T. Anderson},
      doi = {10.1090/pspum/044/840275},
      zblnumber = {0592.53005},
      }
  • [HSi] Go to document 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.
    @article{HSi, mrkey = {0554379},
      author = {Hardt, Robert and Simon, Leon},
      title = {Boundary regularity and embedded solutions for the oriented {P}lateau problem},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {110},
      year = {1979},
      number = {3},
      pages = {439--486},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {49F10 (49F20 53A10)},
      mrnumber = {0554379},
      mrreviewer = {Jo ao Lucas Marqu{ê}s Barbosa},
      zblnumber = {0457.49029},
      doi = {10.2307/1971233},
      }
  • [Ha] Go to document P. Hartman, "Geodesic parallel coordinates in the large," Amer. J. Math., vol. 86, pp. 705-727, 1964.
    @article{Ha, mrkey = {0173222},
      author = {Hartman, Philip},
      title = {Geodesic parallel coordinates in the large},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {86},
      year = {1964},
      pages = {705--727},
      issn = {0002-9327},
      mrclass = {53.75 (30.47)},
      mrnumber = {0173222},
      mrreviewer = {R. Osserman},
      zblnumber = {0128.16105},
      doi = {10.2307/2373154},
      }
  • [Hi] Go to document S. Hildebrandt, "Boundary behavior of minimal surfaces," Arch. Rational Mech. Anal., vol. 35, pp. 47-82, 1969.
    @article{Hi, mrkey = {0248650},
      author = {Hildebrandt, Stefan},
      title = {Boundary behavior of minimal surfaces},
      journal = {Arch. Rational Mech. Anal.},
      fjournal = {Archive for Rational Mechanics and Analysis},
      volume = {35},
      year = {1969},
      pages = {47--82},
      issn = {0003-9527},
      mrclass = {53.04 (30.00)},
      mrnumber = {0248650},
      mrreviewer = {R. Finn},
      zblnumber = {0183.39402},
      doi = {10.1007/BF00248494},
      }
  • [HoKrWe] Go to document 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.
    @article{HoKrWe, mrkey = {1193537},
      author = {Hoffman, David and Wei, Fu Sheng and Karcher, Hermann},
      title = {Adding handles to the helicoid},
      journal = {Bull. Amer. Math. Soc.},
      fjournal = {American Mathematical Society. Bulletin. New Series},
      volume = {29},
      year = {1993},
      number = {1},
      pages = {77--84},
      issn = {0273-0979},
      coden = {BAMOAD},
      mrclass = {53A10 (58E12)},
      mrnumber = {1193537},
      doi = {10.1090/S0273-0979-1993-00401-4},
      zblnumber = {0787.53003},
      }
  • [HoWeWo] Go to document M. Weber, D. Hoffman, and M. Wolf, "An embedded genus-one helicoid," Ann. of Math., vol. 169, iss. 2, pp. 347-448, 2009.
    @article{HoWeWo, mrkey = {2480608},
      author = {Weber, Matthias and Hoffman, David and Wolf, Michael},
      title = {An embedded genus-one helicoid},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {169},
      year = {2009},
      number = {2},
      pages = {347--448},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10 (49Q05)},
      mrnumber = {2480608},
      mrreviewer = {Antonio Alarc{ó}n},
      doi = {10.4007/annals.2009.169.347},
      zblnumber = {1213.49049},
      }
  • [HoMe] Go to document D. Hoffman and W. H. Meeks III, "The strong halfspace theorem for minimal surfaces," Invent. Math., vol. 101, iss. 2, pp. 373-377, 1990.
    @article{HoMe, mrkey = {1062966},
      author = {Hoffman, D. and Meeks, III, W. H.},
      title = {The strong halfspace theorem for minimal surfaces},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {101},
      year = {1990},
      number = {2},
      pages = {373--377},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {53A10},
      mrnumber = {1062966},
      zblnumber = {0722.53054},
      doi = {10.1007/BF01231506},
      }
  • [HoW] Go to document 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.
    @article{HoW, mrkey = {2843239},
      author = {Hoffman, David and White, Brian},
      title = {Sequences of embedded minimal disks whose curvatures blow up on a prescribed subset of a line},
      journal = {Comm. Anal. Geom.},
      fjournal = {Communications in Analysis and Geometry},
      volume = {19},
      year = {2011},
      number = {3},
      pages = {487--502},
      issn = {1019-8385},
      mrclass = {53A10},
      mrnumber = {2843239},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1244.53010},
      doi = {10.4310/CAG.2011.v19.n3.a2},
      }
  • [K] Go to document S. J. Kleene, "A minimal lamination with Cantor set-like singularities," Proc. Amer. Math. Soc., vol. 140, iss. 4, pp. 1423-1436, 2012.
    @article{K, mrkey = {2869127},
      author = {Kleene, Stephen J.},
      title = {A minimal lamination with {C}antor set-like singularities},
      journal = {Proc. Amer. Math. Soc.},
      fjournal = {Proceedings of the American Mathematical Society},
      volume = {140},
      year = {2012},
      number = {4},
      pages = {1423--1436},
      issn = {0002-9939},
      coden = {PAMYAR},
      mrclass = {53A10},
      mrnumber = {2869127},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1239.53006},
      doi = {10.1090/S0002-9939-2011-10971-7},
      }
  • [LaRo] Go to document 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.
    @article{LaRo, mrkey = {0975123},
      author = {Langevin, R{é}mi and Rosenberg, Harold},
      title = {A maximum principle at infinity for minimal surfaces and applications},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {57},
      year = {1988},
      number = {3},
      pages = {819--828},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {53A10 (35J60 49F10 58E12)},
      mrnumber = {0975123},
      mrreviewer = {Marco Rigoli},
      zblnumber = {0667.49024},
      doi = {10.1215/S0012-7094-88-05736-5},
      }
  • [Me1] Go to document 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.
    @article{Me1, mrkey = {2066941},
      author = {Meeks, III, William H.},
      title = {Regularity of the singular set in the {C}olding-{M}inicozzi lamination theorem},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {123},
      year = {2004},
      number = {2},
      pages = {329--334},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {53A10 (49Q05)},
      mrnumber = {2066941},
      mrreviewer = {Man Chun Leung},
      zblnumber = {1086.53005},
      doi = {10.1215/S0012-7094-04-12324-3},
      }
  • [Me2] Go to document 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.
    @article{Me2, mrkey = {2164355},
      author = {Meeks, III, William H.},
      title = {The limit lamination metric for the {C}olding-{M}inicozzi minimal lamination},
      journal = {Illinois J. Math.},
      fjournal = {Illinois Journal of Mathematics},
      volume = {49},
      year = {2005},
      number = {2},
      pages = {645--658},
      issn = {0019-2082},
      coden = {IJMTAW},
      mrclass = {53A10 (53C42)},
      mrnumber = {2164355},
      mrreviewer = {Man Chun Leung},
      url = {http://projecteuclid.org/euclid.ijm/1258138037},
      zblnumber = {1087.53058},
      }
  • [MeP] Go to document 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.
    @incollection{MeP, mrkey = {2195411},
      author = {Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n},
      title = {Conformal properties in classical minimal surface theory},
      booktitle = {Surveys in Differential Geometry. {V}ol. {IX}},
      series = {Surv. Differ. Geom.},
      volume = {IX},
      pages = {275--335},
      publisher = {Int. Press},
      address = {Somerville, MA},
      year = {2004},
      mrclass = {53A10 (49Q05 53C42)},
      mrnumber = {2195411},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1086.53007},
      doi = {10.4310/SDG.2004.v9.n1.a8},
      }
  • [MePRs1] Go to document 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.
    @article{MePRs1, mrkey = {2128712},
      author = {Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n and Ros, Antonio},
      title = {The geometry of minimal surfaces of finite genus. {I}. {C}urvature estimates and quasiperiodicity},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {66},
      year = {2004},
      number = {1},
      pages = {1--45},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53A10 (49Q05 53C42)},
      mrnumber = {2128712},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1068.53012},
      url = {http://projecteuclid.org/euclid.jdg/1090415028},
      }
  • [MePRs2] Go to document 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.
    @article{MePRs2, mrkey = {2096796},
      author = {Meeks, III, William H. and P{é}rez, Joaquin and Ros, Antonio},
      title = {The geometry of minimal surfaces of finite genus. {II}. {N}onexistence of one limit end examples},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {158},
      year = {2004},
      number = {2},
      pages = {323--341},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {53A10 (49Q05 53C42)},
      mrnumber = {2096796},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1070.53003},
      doi = {10.1007/s00222-004-0374-3},
      }
  • [MePRs3] 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.
    @misc{MePRs3,
      author = {Meeks, III, William H. and P{é}rez, Joaquin and Ros, Antonio},
      title = {The geometry of minimal surfaces of finite genus {III}; bounds on the topology and index of classical minimal surfaces},
      note = {preprint},
      sortyear = {2014},
      }
  • [MePRs4] W. H. Meeks III, J. Pérez, and A. Ros, Properly embedded minimal planar domains.
    @misc{MePRs4, KEY={MPR15},
      author = {Meeks, III, William H. and P{é}rez, Joaquin and Ros, Antonio},
      title = {Properly embedded minimal planar domains},
      note = {to appear in {\it Ann. of Math.} {\bf 181} (2015)},
      sortyear = {2015},
      arxiv = {1306.1690},
      }
  • [MePRs5] Go to document 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.
    @article{MePRs5, mrkey = {2629513},
      author = {Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n and Ros, Antonio},
      title = {Limit leaves of an {$H$} lamination are stable},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {84},
      year = {2010},
      number = {1},
      pages = {179--189},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C12},
      mrnumber = {2629513},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {05724752},
      url = {http://projecteuclid.org/euclid.jdg/1271271797},
      }
  • [MePRs6] Go to document 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.
    @article{MePRs6, mrkey = {2273291},
      author = {Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n and Ros, Antonio},
      title = {Liouville-type properties for embedded minimal surfaces},
      journal = {Comm. Anal. Geom.},
      fjournal = {Communications in Analysis and Geometry},
      volume = {14},
      year = {2006},
      number = {4},
      pages = {703--723},
      issn = {1019-8385},
      mrclass = {53A10 (53C42)},
      mrnumber = {2273291},
      mrreviewer = {Fei-Tsen Liang},
      zblnumber = {1117.53009},
      doi = {10.4310/CAG.2006.v14.n4.a5},
      }
  • [MeRo1] Go to document W. H. Meeks III and H. Rosenberg, "The uniqueness of the helicoid," Ann. of Math., vol. 161, iss. 2, pp. 727-758, 2005.
    @article{MeRo1, mrkey = {2153399},
      author = {Meeks, III, William H. and Rosenberg, Harold},
      title = {The uniqueness of the helicoid},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {161},
      year = {2005},
      number = {2},
      pages = {727--758},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10},
      mrnumber = {2153399},
      mrreviewer = {Fei-Tsen Liang},
      doi = {10.4007/annals.2005.161.727},
      zblnumber = {1102.53005},
      }
  • [MeRo2] Go to document W. H. Meeks III and H. Rosenberg, "The minimal lamination closure theorem," Duke Math. J., vol. 133, iss. 3, pp. 467-497, 2006.
    @article{MeRo2, mrkey = {2228460},
      author = {Meeks, III, William H. and Rosenberg, Harold},
      title = {The minimal lamination closure theorem},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {133},
      year = {2006},
      number = {3},
      pages = {467--497},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {53A10 (49Q05 53C42)},
      mrnumber = {2228460},
      mrreviewer = {Fei-Tsen Liang},
      doi = {10.1215/S0012-7094-06-13332-X},
      zblnumber = {1098.53007},
      }
  • [MeWe] Go to document W. H. Meeks III and M. Weber, "Bending the helicoid," Math. Ann., vol. 339, iss. 4, pp. 783-798, 2007.
    @article{MeWe, mrkey = {2341900},
      author = {Meeks, III, William H. and Weber, Matthias},
      title = {Bending the helicoid},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {339},
      year = {2007},
      number = {4},
      pages = {783--798},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {53A10 (49Q05)},
      mrnumber = {2341900},
      mrreviewer = {Stefano Montaldo},
      doi = {10.1007/s00208-007-0120-4},
      zblnumber = {1156.53011},
      }
  • [MeYa1] Go to document 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.
    @article{MeYa1, mrkey = {0670745},
      author = {Meeks, III, William H. and Yau, Shing Tung},
      title = {The classical {P}lateau problem and the topology of three-dimensional manifolds. {T}he embedding of the solution given by {D}ouglas-{M}orrey and an analytic proof of {D}ehn's lemma},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {21},
      year = {1982},
      number = {4},
      pages = {409--442},
      issn = {0040-9383},
      coden = {TPLGAF},
      mrclass = {53A10 (57M35 57N10 58E12)},
      mrnumber = {0670745},
      mrreviewer = {A. J. Tromba},
      doi = {10.1016/0040-9383(82)90021-0},
      zblnumber = {0489.57002},
      }
  • [MeYa2] Go to document 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.
    @article{MeYa2, mrkey = {0645492},
      author = {Meeks, III, William W. and Yau, Shing Tung},
      title = {The existence of embedded minimal surfaces and the problem of uniqueness},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {179},
      year = {1982},
      number = {2},
      pages = {151--168},
      issn = {0025-5874},
      coden = {MAZEAX},
      mrclass = {53C42 (49F10 53A10 58E12)},
      mrnumber = {0645492},
      mrreviewer = {R. Osserman},
      doi = {10.1007/BF01214308},
      zblnumber = {0479.49026},
      }
  • [NO] Go to document K. Nomizu and H. Ozeki, "The existence of complete Riemannian metrics," Proc. Amer. Math. Soc., vol. 12, pp. 889-891, 1961.
    @article{NO, mrkey = {0133785},
      author = {Nomizu, Katsumi and Ozeki, Hideki},
      title = {The existence of complete {R}iemannian metrics},
      journal = {Proc. Amer. Math. Soc.},
      fjournal = {Proceedings of the American Mathematical Society},
      volume = {12},
      year = {1961},
      pages = {889--891},
      issn = {0002-9939},
      mrclass = {53.72},
      mrnumber = {0133785},
      mrreviewer = {S. Kobayashi},
      doi = {10.2307/2034383},
      zblnumber = {0102.16401},
      }
  • [P] J. Pérez, Limits by rescalings of minimal surfaces: Minimal laminations, curvature decay and local pictures.
    @misc{P, key = {P05},
      author = {P{é}rez, J.},
      title = {Limits by rescalings of minimal surfaces: {M}inimal laminations, curvature decay and local pictures},
      note = {notes for the workshop ``Moduli Spaces of Properly Embedded Minimal Surfaces'', American Institute of Mathematics, Palo Alto, California (2005)},
      sortyear = {2005},
      }
  • [Ro] 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.
    @incollection{Ro, mrkey = {2028922},
      author = {Rosenberg, Harold},
      title = {Some recent developments in the theory of minimal surfaces in 3-manifolds},
      booktitle = {24${}^{\rm{o}}$ Col{ó}quio Brasileiro de Matem{á}tica. [24th Brazilian Mathematics Colloquium]},
      series = {Publica\c cões Matemáticas do IMPA. [IMPA Mathematical Publications]},
      publisher = {Instituto de Matemática Pura e Aplicada (IMPA)},
      address = {Rio de Janeiro},
      year = {2003},
      pages = {iv+48},
      isbn = {85-244-0210-5},
      mrclass = {53A10},
      MRNUMBER = {2028922},
      mrreviewer = {Val{é}rio Ramos Batista},
      zblnumber = {1064.53007},
      }
  • [Sc1] 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.
    @incollection{Sc1, mrkey = {0795231},
      author = {Schoen, Richard},
      title = {Estimates for stable minimal surfaces in three-dimensional manifolds},
      booktitle = {Seminar on Minimal Submanifolds},
      series = {Ann. of Math. Stud.},
      volume = {103},
      pages = {111--126},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {1983},
      mrclass = {53C42 (53A10 58E12)},
      mrnumber = {0795231},
      mrreviewer = {Richard H. Escobales, Jr.},
      zblnumber = {0532.53042},
      }
  • [Sc2] Go to document R. M. Schoen, "Uniqueness, symmetry, and embeddedness of minimal surfaces," J. Differential Geom., vol. 18, iss. 4, pp. 791-809 (1984), 1983.
    @article{Sc2, mrkey = {0730928},
      author = {Schoen, Richard M.},
      title = {Uniqueness, symmetry, and embeddedness of minimal surfaces},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {18},
      year = {1983},
      number = {4},
      pages = {791--809 (1984)},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53A10 (58E12)},
      mrnumber = {0730928},
      mrreviewer = {V. M. Miklyukov},
      url = {http://projecteuclid.org/euclid.jdg/1214438183},
      zblnumber = {0575.53037},
      }
  • [ShTa1] 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.
    @incollection{ShTa1, mrkey = {1040533},
      author = {Shiohama, K. and Tanaka, M.},
      title = {An isoperimetric problem for infinitely connected complete open surfaces},
      booktitle = {Geometry of Manifolds},
      venue = {{M}atsumoto, 1988},
      series = {Perspect. Math.},
      volume = {8},
      pages = {317--343},
      publisher = {Academic Press},
      address = {Boston},
      year = {1989},
      mrclass = {53C20},
      mrnumber = {1040533},
      mrreviewer = {Sheila Carter},
      zblnumber = {0697.53040},
      }
  • [ShTa2] 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.
    @incollection{ShTa2, mrkey = {1274955},
      author = {Shiohama, Katsuhiro and Tanaka, Minoru},
      title = {The length function of geodesic parallel circles},
      booktitle = {Progress in Differential Geometry},
      series = {Adv. Stud. Pure Math.},
      volume = {22},
      pages = {299--308},
      publisher = {Math. Soc. Japan},
      address = {Tokyo},
      year = {1993},
      mrclass = {53C22},
      mrnumber = {1274955},
      mrreviewer = {Sheila Carter},
      zblnumber = {0799.53052},
      }
  • [So] Go to document B. Solomon, "On foliations of ${\bf R}^{n+1}$ by minimal hypersurfaces," Comment. Math. Helv., vol. 61, iss. 1, pp. 67-83, 1986.
    @article{So, mrkey = {0847521},
      author = {Solomon, Bruce},
      title = {On foliations of {${\bf R}\sp {n+1}$} by minimal hypersurfaces},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {61},
      year = {1986},
      number = {1},
      pages = {67--83},
      issn = {0010-2571},
      coden = {COMHAX},
      mrclass = {49F10 (53C12 53C42 57R30)},
      mrnumber = {0847521},
      mrreviewer = {Friedrich Sauvigny},
      doi = {10.1007/BF02621903},
      zblnumber = {0601.53025},
      }

Authors

Tobias H. Colding

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY

Current address:

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307 William P. Minicozzi II

Department of Mathematics, Johns Hopkins University, Baltimore, MD

Current address:

Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307