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

Tobias H. Colding; William P. Minicozzi II

doi:https://doi.org/10.4007/annals.2015.181.1.1

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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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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}, }
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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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}, }

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

Abstract

Authors