Abstract
In 1997, Collin proved that any properly embedded minimal surface in $\mathbb{R}^3$ with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of López and Ros, catenoids are the only nonplanar, nonsimply connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ of finite topology. In 2005, Meeks and Rosenberg proved that the only simply connected, properly embedded minimal surfaces in $\mathbb{R}^3$ are planes and helicoids. Around 1860, Riemann defined a one-parameter family of periodic, infinite topology, properly embedded, minimal planar domains $\mathcal{R}_t$ in $\mathbb{R}^3$, $t \in (0,\infty)$. These surfaces are called the Riemann minimal examples, and the family $\{ \mathcal{R}_t\} _t$ has natural limits being a vertical catenoid as $t\to~0$ and a vertical helicoid as $t\to\infty$. In this paper we complete the classification of properly embedded, minimal planar domains in $\mathbb{R}^3$ by proving that the only connected examples with infinite topology are the Riemann minimal examples. We also prove that the limit ends of Riemann minimal examples are model surfaces for the limit ends of properly embedded minimal surfaces $M\subset\mathbb{R}^3$ of finite genus and infinite topology, in the sense that such an $M$ has two limit ends, each of which has a representative that is naturally asymptotic to a limit end representative of a Riemann minimal example with the same associated flux vector.
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MRCLASS = {53A10},
MRNUMBER = {1085145},
MRREVIEWER = {Cun-Jin Sun},
URL = {http://projecteuclid.org/euclid.jdg/1214446040},
ZBLNUMBER={0719.53004},
} -
[me30]
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 {me30, 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},
} -
@misc{mpe3,
author = {Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n},
TITLE = {Embedded minimal surfaces of finite topology},
URL={http://wdb.ugr.es/~jperez/publications-by-joaquin-perez/},
SORTYEAR={2019},
} -
[mpe1]
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, pp. 275-335.
@incollection {mpe1, 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}},
PAGES = {275--335},
PUBLISHER = {Int. Press},
ADDRESS={Somerville, MA},
YEAR = {2004},
MRCLASS = {53A10 (49Q05 53C42)},
MRNUMBER = {2195411},
MRREVIEWER = {Fei-Tsen Liang},
DOI = {10.4310/SDG.2004.v9.n1.a8},
ZBLNUMBER={1086.53007},
} -
[mpr8]
W. H. Meeks III, J. Pérez, and A. Ros, Bounds on the topology and index of classical minimal surfaces.
@misc{mpr8,
author={Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n and Ros, Antonio},
title={Bounds on the topology and index of classical minimal surfaces},
URL={http://wdb.ugr.es/~jperez/publications-by-joaquin-perez/},
SORTYEAR={2017},
} -
@misc{mpr9,
author={Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n and Ros, Antonio},
title={The embedded {C}alabi-{Y}au conjectures for finite genus},
URL={http://wdb.ugr.es/~jperez/publications-by-joaquin-perez/},
SORTYEAR={2018},
} -
[mpr1]
W. H. Meeks III, J. Pérez, and A. Ros, "Uniqueness of the Riemann minimal examples," Invent. Math., vol. 133, iss. 1, pp. 107-132, 1998.
@article {mpr1, MRKEY = {1626477},
AUTHOR = {Meeks, III, William H. and P{é}rez, Joaqu{\'ı}n and Ros, Antonio},
TITLE = {Uniqueness of the {R}iemann minimal examples},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {133},
YEAR = {1998},
NUMBER = {1},
PAGES = {107--132},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53A10},
MRNUMBER = {1626477},
MRREVIEWER = {Kichoon Yang},
DOI = {10.1007/s002220050241},
ZBLNUMBER={0916.53004},
} -
[mpr3]
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 {mpr3, 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},
URL = {http://projecteuclid.org/euclid.jdg/1090415028},
ZBLNUMBER={1068.53012},
} -
[mpr4]
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 {mpr4, 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},
DOI = {10.1007/s00222-004-0374-3},
ZBLNUMBER={1070.53003},
} -
[mr1]
W. H. Meeks III and H. Rosenberg, "The maximum principle at infinity for minimal surfaces in flat three manifolds," Comment. Math. Helv., vol. 65, iss. 2, pp. 255-270, 1990.
@article {mr1, MRKEY = {1057243},
AUTHOR = {Meeks, III, William H. and Rosenberg, Harold},
TITLE = {The maximum principle at infinity for minimal surfaces in flat three manifolds},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {65},
YEAR = {1990},
NUMBER = {2},
PAGES = {255--270},
ISSN = {0010-2571},
CODEN = {COMHAX},
MRCLASS = {53A10 (53C42)},
MRNUMBER = {1057243},
MRREVIEWER = {Xiao Wei Peng},
DOI = {10.1007/BF02566606},
ZBLNUMBER={0713.53008},
} -
[mr8]
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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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},
} -
[meu1] J. B. Meusnier, "Mémoire sur la courbure des surfaces," Mém. Mathém. Phys. Acad. Sci. Paris, prés. par div. Savans, vol. 10, pp. 477-510, 1785.
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author={J. B. Meusnier},
title={Mémoire sur la courbure des surfaces},
journal={Mém. Mathém. Phys. Acad. Sci. Paris, prés. par div. Savans},
volume={10},
year={1785},
pages={477--510},
note={presented in 1776},
} -
[mro1]
S. Montiel and A. Ros, "Schrödinger operators associated to a holomorphic map," in Global Differential Geometry and Global Analysis, New York: Springer-Verlag, 1991, vol. 1481, pp. 147-174.
@incollection {mro1, MRKEY = {1178529},
AUTHOR = {Montiel, Sebasti{á}n and Ros, Antonio},
TITLE = {Schrödinger operators associated to a holomorphic map},
BOOKTITLE = {Global Differential Geometry and Global Analysis},
VENUE={{B}erlin, 1990},
SERIES = {Lecture Notes in Math.},
VOLUME = {1481},
PAGES = {147--174},
PUBLISHER = {Springer-Verlag},
YEAR = {1991},
MRCLASS = {58E11 (53A10 58G26)},
MRNUMBER = {1178529},
MRREVIEWER = {Giorgio Valli},
DOI = {10.1007/BFb0083639},
ADDRESS = {New York},
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} -
[os3]
R. Osserman, "Global properties of minimal surfaces in $E^{3}$ and $E^{n}$," Ann. of Math., vol. 80, pp. 340-364, 1964.
@article {os3, MRKEY = {0179701},
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TITLE = {Global properties of minimal surfaces in {$E\sp{3}$} and {$E\sp{n}$}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {80},
YEAR = {1964},
PAGES = {340--364},
ISSN = {0003-486X},
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[os1] R. Osserman, A Survey of Minimal Surfaces, Second ed., New York: Dover Publications, 1986.
@book {os1, MRKEY = {0852409},
AUTHOR = {Osserman, Robert},
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EDITION = {Second},
PUBLISHER = {Dover Publications},
ADDRESS={New York},
YEAR = {1986},
PAGES = {vi+207},
ISBN = {0-486-64998-9},
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[perez3]
J. Pérez, "On singly-periodic minimal surfaces with planar ends," Trans. Amer. Math. Soc., vol. 349, iss. 6, pp. 2371-2389, 1997.
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AUTHOR = {P{é}rez, Joaqu{\'ı}n},
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JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
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YEAR = {1997},
NUMBER = {6},
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ISSN = {0002-9947},
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DOI = {10.1090/S0002-9947-97-01911-9},
ZBLNUMBER={0882.53007},
} -
[pro4]
J. Pérez and A. Ros, "The space of properly embedded minimal surfaces with finite total curvature," Indiana Univ. Math. J., vol. 45, iss. 1, pp. 177-204, 1996.
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AUTHOR = {P{é}rez, Joaqu{\'ı}n and Ros, Antonio},
TITLE = {The space of properly embedded minimal surfaces with finite total curvature},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {45},
YEAR = {1996},
NUMBER = {1},
PAGES = {177--204},
ISSN = {0022-2518},
CODEN = {IUMJAB},
MRCLASS = {58D27 (53A10 58E12)},
MRNUMBER = {1406689},
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DOI = {10.1512/iumj.1996.45.2053},
ZBLNUMBER={0864.53008},
} -
[ri1] B. Riemann, Ouevres Mathématiques de Riemann, Paris: Gauthiers-Villars, 1898.
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author={B. Riemann},
title={Ouevres Mathématiques de {R}iemann},
publisher={Gauthiers-Villars},
address={Paris},
year={1898},
JFM = {29.0009.02},
} -
[ri2] B. Riemann, "Über die Fläche vom kleinsten Inhalt bei gegebener Begrenzung," Abh. Königl, d. Wiss. Göttingen, Mathem. Cl., vol. 13, pp. 3-52, 1867.
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author={B. Riemann},
title={Über die {F}läche vom kleinsten {I}nhalt bei gegebener {B}egrenzung},
journal={Abh. Königl, d. Wiss. Göttingen, Mathem. Cl.},
volume={13},
year={1867},
pages={3--52},
EDITOR={K. Hattendorf},
JFMNUMBER={01.0218.01},
} -
[sc3] 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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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},
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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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AUTHOR = {Schoen, Richard M.},
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JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {18},
YEAR = {1983},
NUMBER = {4},
PAGES = {791--809 (1984)},
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[SeWi]
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M. Shiffman, "On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes," Ann. of Math., vol. 63, pp. 77-90, 1956.
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[weik1]
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[wh1]
B. White, "The space of $m$-dimensional surfaces that are stationary for a parametric elliptic functional," Indiana Univ. Math. J., vol. 36, iss. 3, pp. 567-602, 1987.
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ZBLNUMBER={0770.58005},
}