Abstract
In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $\mathbb{R}^3$ is at least $2\pi^2$. We prove this conjecture using the min-max theory of minimal surfaces.
-
[allard-almgren]
W. K. Allard and F. J. Almgren Jr., "The structure of stationary one dimensional varifolds with positive density," Invent. Math., vol. 34, iss. 2, pp. 83-97, 1976.
@article {allard-almgren, MRKEY = {0425741},
AUTHOR = {Allard, W. K. and Almgren, Jr., F. J.},
TITLE = {The structure of stationary one dimensional varifolds with positive density},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {34},
YEAR = {1976},
NUMBER = {2},
PAGES = {83--97},
ISSN = {0020-9910},
MRCLASS = {49F20 (58E15)},
MRNUMBER = {0425741},
MRREVIEWER = {J. E. Brothers},
DOI = {10.1007/BF01425476},
ZBLNUMBER = {0339.49020},
} -
[almgren]
F. J. Almgren Jr., "The homotopy groups of the integral cycle groups," Topology, vol. 1, pp. 257-299, 1962.
@article {almgren, MRKEY = {0146835},
AUTHOR = {Almgren, Jr., Frederick Justin},
TITLE = {The homotopy groups of the integral cycle groups},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {1},
YEAR = {1962},
PAGES = {257--299},
ISSN = {0040-9383},
MRCLASS = {55.45 (55.42)},
MRNUMBER = {0146835},
MRREVIEWER = {W. W. Fleming},
DOI = {10.1016/0040-9383(62)90016-2},
ZBLNUMBER = {0118.18503},
} -
[almgren-varifolds] F. J. Almgren Jr., The theory of varifolds, 1965.
@misc{almgren-varifolds,
author = {Almgren, Jr., Frederick Justin},
TITLE = {The theory of varifolds},
NOTE={mimeographed notes, Princeton},
YEAR={1965},
} -
[almgren66]
F. J. Almgren Jr., "Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem," Ann. of Math., vol. 84, pp. 277-292, 1966.
@article {almgren66, MRKEY = {0200816},
AUTHOR = {Almgren, Jr., Frederick Justin},
TITLE = {Some interior regularity theorems for minimal surfaces and an extension of {B}ernstein's theorem},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {84},
YEAR = {1966},
PAGES = {277--292},
ISSN = {0003-486X},
MRCLASS = {53.04},
MRNUMBER = {0200816},
MRREVIEWER = {W. P. Ziemer},
DOI = {10.2307/1970520},
ZBLNUMBER = {0146.11905},
} -
[ammann]
B. Ammann, "The Willmore conjecture for immersed tori with small curvature integral," Manuscripta Math., vol. 101, iss. 1, pp. 1-22, 2000.
@article {ammann, MRKEY = {1737221},
AUTHOR = {Ammann, Bernd},
TITLE = {The {W}illmore conjecture for immersed tori with small curvature integral},
JOURNAL = {Manuscripta Math.},
FJOURNAL = {Manuscripta Mathematica},
VOLUME = {101},
YEAR = {2000},
NUMBER = {1},
PAGES = {1--22},
ISSN = {0025-2611},
CODEN = {MSMHB2},
MRCLASS = {53C42 (53A07 53A10)},
MRNUMBER = {1737221},
MRREVIEWER = {Udo Hertrich-Jeromin},
DOI = {10.1007/s002290050001},
ZBLNUMBER = {0959.53032},
} -
[bauer-kuwert]
M. Bauer and E. Kuwert, "Existence of minimizing Willmore surfaces of prescribed genus," Int. Math. Res. Not., vol. 2003, iss. 10, pp. 553-576, 2003.
@article {bauer-kuwert, MRKEY = {1941840},
AUTHOR = {Bauer, Matthias and Kuwert, Ernst},
TITLE = {Existence of minimizing {W}illmore surfaces of prescribed genus},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2003},
NUMBER = {10},
PAGES = {553--576},
ISSN = {1073-7928},
MRCLASS = {53C42 (53A10)},
MRNUMBER = {1941840},
MRREVIEWER = {Shu-Cheng Chang},
DOI = {10.1155/S1073792803208072},
VOLUME = {2003},
ZBLNUMBER = {1029.53073},
} -
[bensimon]
M. Mutz and D. Benisimon, "Observation of toroidal vesicle," Phys. Rev. A., vol. 43, pp. 4525-4527, 1991.
@article{bensimon,
author={Mutz, M. and Benisimon, D.},
TITLE={Observation of toroidal vesicle},
JOURNAL={Phys. Rev. A.},
VOLUME={43},
YEAR={1991},
PAGES={4525--4527},
DOI = {10.1103/PhysRevA.43.4525},
} -
[blaschke] W. Blaschke, Vorlesungen Über Differentialgeometrie III, Berlin: Springer-Verlag, 1929.
@book{blaschke,
author={Blaschke, W.},
TITLE={Vorlesungen Über Differentialgeometrie III},
PUBLISHER={Springer-Verlag},
ADDRESS={Berlin},
YEAR={1929},
JFMNUMBER = {55.0422.01},
} -
[bryant2]
R. L. Bryant, "A duality theorem for Willmore surfaces," J. Differential Geom., vol. 20, iss. 1, pp. 23-53, 1984.
@article {bryant2, MRKEY = {0772125},
AUTHOR = {Bryant, Robert L.},
TITLE = {A duality theorem for {W}illmore surfaces},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {20},
YEAR = {1984},
NUMBER = {1},
PAGES = {23--53},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {58E12 (53C42)},
MRNUMBER = {0772125},
MRREVIEWER = {Akito Futaki},
URL = {http://projecteuclid.org/euclid.jdg/1214438991},
ZBLNUMBER = {0555.53002},
} -
[bryant]
R. L. Bryant, "Surfaces in conformal geometry," in The Mathematical Heritage of Hermann Weyl, Providence, RI: Amer. Math. Soc., 1988, vol. 48, pp. 227-240.
@incollection {bryant, MRKEY = {0974338},
AUTHOR = {Bryant, Robert L.},
TITLE = {Surfaces in conformal geometry},
BOOKTITLE = {The Mathematical Heritage of {H}ermann {W}eyl},
VENUE={{D}urham, {NC},
1987},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {48},
PAGES = {227--240},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1988},
MRCLASS = {53C42 (32C10 53A10 58E12)},
MRNUMBER = {0974338},
MRREVIEWER = {John M. Lee},
ZBLNUMBER = {0654.53010},
DOI = {10.1090/pspum/048},
} -
[chen]
B. Chen, "On the total curvature of immersed manifolds. VI. Submanifolds of finite type and their applications," Bull. Inst. Math. Acad. Sinica, vol. 11, iss. 3, pp. 309-328, 1983.
@article {chen, MRKEY = {0726979},
AUTHOR = {Chen, Bang-Yen},
TITLE = {On the total curvature of immersed manifolds. {VI}. {S}ubmanifolds of finite type and their applications},
JOURNAL = {Bull. Inst. Math. Acad. Sinica},
FJOURNAL = {Bulletin of the Institute of Mathematics. Academia Sinica},
VOLUME = {11},
YEAR = {1983},
NUMBER = {3},
PAGES = {309--328},
ISSN = {0304-9825},
CODEN = {BIMSDG},
MRCLASS = {53C40 (53C42)},
MRNUMBER = {0726979},
MRREVIEWER = {Shoshichi Kobayashi},
ZBLNUMBER = {0514.53049},
URL = {http://w3.math.sinica.edu.tw/bulletin/bulletin_id_a.jsp?bid=MTk4MzMwMg==},
} -
[choi-schoen]
H. I. Choi and R. Schoen, "The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature," Invent. Math., vol. 81, iss. 3, pp. 387-394, 1985.
@article {choi-schoen, MRKEY = {0807063},
AUTHOR = {Choi, Hyeong In and Schoen, Richard},
TITLE = {The space of minimal embeddings of a surface into a three-dimensional manifold of positive {R}icci curvature},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {81},
YEAR = {1985},
NUMBER = {3},
PAGES = {387--394},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58E12 (53C42 58D10)},
MRNUMBER = {0807063},
MRREVIEWER = {J. Eells},
DOI = {10.1007/BF01388577},
} -
[colding-delellis] T. H. Colding and C. De Lellis, "The min-max construction of minimal surfaces," in Surveys in Differential Geometry, Vol. VIII, Int. Press, Somerville, MA, 2003, vol. VIII, pp. 75-107.
@incollection {colding-delellis, MRKEY = {2039986},
AUTHOR = {Colding, Tobias H. and De Lellis, Camillo},
TITLE = {The min-max construction of minimal surfaces},
BOOKTITLE = {Surveys in Differential Geometry, {V}ol. {VIII}},
VENUE={{B}oston, {MA},
2002},
SERIES = {Surv. Differ. Geom.},
VOLUME={VIII},
PAGES = {75--107},
PUBLISHER = {Int. Press, Somerville, MA},
YEAR = {2003},
MRCLASS = {53A10 (49Q05 53C42)},
MRNUMBER = {2039986},
MRREVIEWER = {Fei-Tsen Liang},
ZBLNUMBER = {1051.53052},
} -
[evans] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Boca Raton, FL: CRC Press, 1992.
@book {evans, MRKEY = {1158660},
AUTHOR = {Evans, Lawrence C. and Gariepy, Ronald F.},
TITLE = {Measure Theory and Fine Properties of Functions},
SERIES = {Stud. Adv. Math.},
PUBLISHER = {CRC Press},
ADDRESS = {Boca Raton, FL},
YEAR = {1992},
PAGES = {viii+268},
ISBN = {0-8493-7157-0},
MRCLASS = {28-02 (26-02 26Bxx 46E35)},
MRNUMBER = {1158660},
MRREVIEWER = {R. G. Bartle},
ZBLNUMBER = {0804.28001},
} -
[germain] S. Germain, Recherches sur la théorie des surfaces élastiques, 1921.
@misc{germain,
author={Germain, S.},
TITLE={Recherches sur la théorie des surfaces élastiques},
NOTE={Paris},
YEAR={1921},
} -
[heintze-karcher]
E. Heintze and H. Karcher, "A general comparison theorem with applications to volume estimates for submanifolds," Ann. Sci. École Norm. Sup., vol. 11, iss. 4, pp. 451-470, 1978.
@article {heintze-karcher, MRKEY = {0533065},
AUTHOR = {Heintze, Ernst and Karcher, Hermann},
TITLE = {A general comparison theorem with applications to volume estimates for submanifolds},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {11},
YEAR = {1978},
NUMBER = {4},
PAGES = {451--470},
ISSN = {0012-9593},
CODEN = {ENAQAF},
MRCLASS = {53C40 (58E10)},
MRNUMBER = {0533065},
MRREVIEWER = {Hubert Gollek},
URL = {http://www.numdam.org/item?id=ASENS_1978_4_11_4_451_0},
ZBLNUMBER = {0416.53027},
} -
[helfrich] W. Helfrich, "Elastic properties of lipid bilayers: Theory and possible experiments," Z. Naturforsch., vol. 28, pp. 693-703, 1973.
@article{helfrich,
author={Helfrich, W.},
TITLE={Elastic properties of lipid bilayers: {T}heory and possible experiments},
JOURNAL={Z. Naturforsch.},
VOLUME={28},
YEAR={1973},
PAGES={693--703},
} -
[pinkall2]
U. Hertrich-Jeromin and U. Pinkall, "Ein Beweis der Willmoreschen Vermutung für Kanaltori," J. Reine Angew. Math., vol. 430, pp. 21-34, 1992.
@article {pinkall2, MRKEY = {1172905},
AUTHOR = {Hertrich-Jeromin, U. and Pinkall, U.},
TITLE = {Ein {B}eweis der {W}illmoreschen {V}ermutung für {K}analtori},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
VOLUME = {430},
YEAR = {1992},
PAGES = {21--34},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {53C40 (53A05)},
MRNUMBER = {1172905},
DOI = {10.1515/crll.1992.430.21},
ZBLNUMBER = {0749.53040},
} -
[kuhnel-pinkall]
W. Kühnel and U. Pinkall, "On total mean curvatures," Quart. J. Math. Oxford Ser., vol. 37, iss. 148, pp. 437-447, 1986.
@article {kuhnel-pinkall, MRKEY = {0868618},
AUTHOR = {K{ü}hnel, Wolfgang and Pinkall, Ulrich},
TITLE = {On total mean curvatures},
JOURNAL = {Quart. J. Math. Oxford Ser.},
FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series},
VOLUME = {37},
YEAR = {1986},
NUMBER = {148},
PAGES = {437--447},
ISSN = {0033-5606},
CODEN = {QJMAAT},
MRCLASS = {53C65 (53A07)},
MRNUMBER = {0868618},
MRREVIEWER = {Ralph Howard},
DOI = {10.1093/qmath/37.4.437},
ZBLNUMBER = {0627.53044},
} -
[kusner]
R. Kusner, "Conformal geometry and complete minimal surfaces," Bull. Amer. Math. Soc., vol. 17, iss. 2, pp. 291-295, 1987.
@article {kusner, MRKEY = {0903735},
AUTHOR = {Kusner, Rob},
TITLE = {Conformal geometry and complete minimal surfaces},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {17},
YEAR = {1987},
NUMBER = {2},
PAGES = {291--295},
ISSN = {0273-0979},
CODEN = {BAMOAD},
MRCLASS = {53A10 (49F10 57R42)},
MRNUMBER = {0903735},
MRREVIEWER = {T. Banchoff},
DOI = {10.1090/S0273-0979-1987-15564-9},
ZBLNUMBER = {0634.53004},
} -
[kusner89]
R. Kusner, "Comparison surfaces for the Willmore problem," Pacific J. Math., vol. 138, iss. 2, pp. 317-345, 1989.
@article {kusner89, MRKEY = {0996204},
AUTHOR = {Kusner, Rob},
TITLE = {Comparison surfaces for the {W}illmore problem},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {138},
YEAR = {1989},
NUMBER = {2},
PAGES = {317--345},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {53A10 (53A30 53C42 58E12)},
MRNUMBER = {0996204},
MRREVIEWER = {Joel L. Weiner},
DOI = {10.2140/pjm.1989.138.317},
ZBLNUMBER = {0643.53044},
} -
[kusner96] R. Kusner, "Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral," in Elliptic and Parabolic Methods in Geometry, Wellesley, MA: A K Peters, 1996, pp. 67-72.
@incollection {kusner96, MRKEY = {1417949},
AUTHOR = {Kusner, Rob},
TITLE = {Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral},
BOOKTITLE = {Elliptic and Parabolic Methods in Geometry},
VENUE={{M}inneapolis, {MN},
1994},
PAGES = {67--72},
PUBLISHER = {A K Peters},
ADDRESS = {Wellesley, MA},
YEAR = {1996},
MRCLASS = {53A10 (58E12)},
MRNUMBER = {1417949},
MRREVIEWER = {Udo Hertrich-Jeromin},
ZBLNUMBER = {0874.49038},
} -
[kuwert-schatzle]
E. Kuwert and R. Schätzle, "Removability of point singularities of Willmore surfaces," Ann. of Math., vol. 160, iss. 1, pp. 315-357, 2004.
@article {kuwert-schatzle, MRKEY = {2119722},
AUTHOR = {Kuwert, Ernst and Sch{ä}tzle, Reiner},
TITLE = {Removability of point singularities of {W}illmore surfaces},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {160},
YEAR = {2004},
NUMBER = {1},
PAGES = {315--357},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {58E12 (53C44)},
MRNUMBER = {2119722},
MRREVIEWER = {Shu-Cheng Chang},
DOI = {10.4007/annals.2004.160.315},
ZBLNUMBER = {1078.53007},
} -
[kuwert-li-schatzle]
E. Kuwert, Y. Li, and R. Schätzle, "The large genus limit of the infimum of the Willmore energy," Amer. J. Math., vol. 132, iss. 1, pp. 37-51, 2010.
@article {kuwert-li-schatzle, MRKEY = {2597505},
AUTHOR = {Kuwert, Ernst and Li, Yuxiang and Sch{ä}tzle, Reiner},
TITLE = {The large genus limit of the infimum of the {W}illmore energy},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {132},
YEAR = {2010},
NUMBER = {1},
PAGES = {37--51},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {58E12 (53C42)},
MRNUMBER = {2597505},
MRREVIEWER = {Andreas Gastel},
DOI = {10.1353/ajm.0.0100},
ZBLNUMBER = {1188.53057},
} -
[langer-singer]
J. Langer and D. Singer, "Curves in the hyperbolic plane and mean curvature of tori in $3$-space," Bull. London Math. Soc., vol. 16, iss. 5, pp. 531-534, 1984.
@article {langer-singer, MRKEY = {0751827},
AUTHOR = {Langer, Joel and Singer, David},
TITLE = {Curves in the hyperbolic plane and mean curvature of tori in {$3$}-space},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical Society},
VOLUME = {16},
YEAR = {1984},
NUMBER = {5},
PAGES = {531--534},
ISSN = {0024-6093},
CODEN = {LMSBBT},
MRCLASS = {53A05 (53A35)},
MRNUMBER = {0751827},
MRREVIEWER = {R. Osserman},
DOI = {10.1112/blms/16.5.531},
ZBLNUMBER = {0554.53014},
} -
[langevin-rosenberg]
R. Langevin and H. Rosenberg, "On curvature integrals and knots," Topology, vol. 15, iss. 4, pp. 405-416, 1976.
@article {langevin-rosenberg, MRKEY = {0440562},
AUTHOR = {Langevin, R{é}mi and Rosenberg, Harold},
TITLE = {On curvature integrals and knots},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {15},
YEAR = {1976},
NUMBER = {4},
PAGES = {405--416},
ISSN = {0040-9383},
MRCLASS = {57C45 (57D70)},
MRNUMBER = {0440562},
MRREVIEWER = {Louis H. Kauffman},
DOI = {10.1016/0040-9383(76)90034-3},
ZBLNUMBER = {0341.53037},
} -
[lawson70]
B. H. Lawson Jr., "Complete minimal surfaces in $S^{3}$," Ann. of Math., vol. 92, pp. 335-374, 1970.
@article {lawson70, MRKEY = {0270280},
AUTHOR = {Lawson, Jr., H. Blaine},
TITLE = {Complete minimal surfaces in {$S\sp{3}$}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {92},
YEAR = {1970},
PAGES = {335--374},
ISSN = {0003-486X},
MRCLASS = {53.04},
MRNUMBER = {0270280},
MRREVIEWER = {C. S. Weaver},
DOI = {10.2307/1970625},
ZBLNUMBER = {0205.52001},
} -
[li-yau]
P. Li and S. T. Yau, "A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces," Invent. Math., vol. 69, iss. 2, pp. 269-291, 1982.
@article {li-yau, MRKEY = {0674407},
AUTHOR = {Li, Peter and Yau, Shing Tung},
TITLE = {A new conformal invariant and its applications to the {W}illmore conjecture and the first eigenvalue of compact surfaces},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {69},
YEAR = {1982},
NUMBER = {2},
PAGES = {269--291},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53C40 (53C21 58G25)},
MRNUMBER = {0674407},
MRREVIEWER = {Yu. Burago},
DOI = {10.1007/BF01399507},
ZBLNUMBER = {0503.53042},
} -
[massey] W. S. Massey, A Basic Course in Algebraic Topology, New York: Springer-Verlag, 1991, vol. 127.
@book {massey, MRKEY = {1095046},
AUTHOR = {Massey, William S.},
TITLE = {A Basic Course in Algebraic Topology},
SERIES = {Grad. Texts in Math.},
VOLUME = {127},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1991},
PAGES = {xvi+428},
ISBN = {0-387-97430-X},
MRCLASS = {55-02 (57-02)},
MRNUMBER = {1095046},
MRREVIEWER = {Jo{ž}e Vrabec},
ZBLNUMBER = {0725.55001},
} -
[bensimon2]
X. Michalet and D. Bensimon, "Vesicles of toroidal topology: observed morphology and shape transformations," J. Phys. II France, vol. 5, pp. 263-287, 1995.
@article{bensimon2,
author={Michalet, X. and Bensimon, D.},
TITLE={Vesicles of toroidal topology: observed morphology and shape transformations},
JOURNAL={J. Phys. II France},
VOLUME={5},
YEAR={1995},
PAGES={263--287},
DOI = {10.1051/jp2:1995128},
} -
[montiel-ros]
S. Montiel and A. Ros, "Minimal immersions of surfaces by the first eigenfunctions and conformal area," Invent. Math., vol. 83, iss. 1, pp. 153-166, 1985.
@article {montiel-ros, MRKEY = {0813585},
AUTHOR = {Montiel, Sebasti{á}n and Ros, Antonio},
TITLE = {Minimal immersions of surfaces by the first eigenfunctions and conformal area},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {83},
YEAR = {1985},
NUMBER = {1},
PAGES = {153--166},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53C42 (58G25)},
MRNUMBER = {0813585},
MRREVIEWER = {Carolyn Gordon},
DOI = {10.1007/BF01388756},
ZBLNUMBER = {0584.53026},
} -
@article {pinkall, MRKEY = {0799274},
AUTHOR = {Pinkall, U.},
TITLE = {Hopf tori in {$S\sp 3$}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {81},
YEAR = {1985},
NUMBER = {2},
PAGES = {379--386},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53C42 (53A10 57R40)},
MRNUMBER = {0799274},
MRREVIEWER = {Sheila Carter},
DOI = {10.1007/BF01389060},
ZBLNUMBER = {0585.53051},
} -
[pitts] J. T. Pitts, Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, Princeton, N.J.: Princeton Univ. Press, 1981, vol. 27.
@book {pitts, MRKEY = {0626027},
AUTHOR = {Pitts, Jon T.},
TITLE = {Existence and Regularity of Minimal Surfaces on {R}iemannian Manifolds},
SERIES = {Math. Notes},
VOLUME = {27},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1981},
PAGES = {iv+330},
ISBN = {0-691-08290-1},
MRCLASS = {49F22 (53C42)},
MRNUMBER = {0626027},
MRREVIEWER = {J. E. Brothers},
ZBLNUMBER = {0462.58003},
} -
[poisson] S. D. Poisson, Mémoire sur les Surfaces Élastiques, Inst. de France, 1812.
@book{poisson,
author={Poisson, S. D.},
TITLE={Mémoire sur les Surfaces Élastiques},
SERIES={Mem. Cl. Sci. Math. Phys.},
PUBLISHER={Inst. de France},
YEAR={1812},
PAGES={167--225},
} -
[riviere]
T. Rivière, "Analysis aspects of Willmore surfaces," Invent. Math., vol. 174, iss. 1, pp. 1-45, 2008.
@article {riviere, MRKEY = {2430975},
AUTHOR = {Rivi{è}re, Tristan},
TITLE = {Analysis aspects of {W}illmore surfaces},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {174},
YEAR = {2008},
NUMBER = {1},
PAGES = {1--45},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53C42 (35J60 53C45)},
MRNUMBER = {2430975},
MRREVIEWER = {Andreas Gastel},
DOI = {10.1007/s00222-008-0129-7},
ZBLNUMBER = {1155.53031},
} -
[ros]
A. Ros, "The Willmore conjecture in the real projective space," Math. Res. Lett., vol. 6, iss. 5-6, pp. 487-493, 1999.
@article {ros, MRKEY = {1739208},
AUTHOR = {Ros, Antonio},
TITLE = {The {W}illmore conjecture in the real projective space},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {6},
YEAR = {1999},
NUMBER = {5-6},
PAGES = {487--493},
ISSN = {1073-2780},
MRCLASS = {53A10 (53C42)},
MRNUMBER = {1739208},
MRREVIEWER = {Wolfgang K{ü}hnel},
DOI = {10.4310/MRL.1999.v6.n5.a2},
ZBLNUMBER = {0951.53044},
} -
[ros2001]
A. Ros, "The isoperimetric and Willmore problems," in Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Providence, RI: Amer. Math. Soc., 2001, vol. 288, pp. 149-161.
@incollection {ros2001, MRKEY = {1871006},
AUTHOR = {Ros, Antonio},
TITLE = {The isoperimetric and {W}illmore problems},
BOOKTITLE = {Global Differential Geometry: The Mathematical Legacy of {A}lfred {G}ray},
VENUE={{B}ilbao, 2000},
SERIES = {Contemp. Math.},
VOLUME = {288},
PAGES = {149--161},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2001},
MRCLASS = {53A10 (53C42)},
MRNUMBER = {1871006},
MRREVIEWER = {Ildefonso Castro},
DOI = {10.1090/conm/288/04823},
ZBLNUMBER = {1005.53054},
} -
[schoen-simon]
R. Schoen and L. Simon, "Regularity of stable minimal hypersurfaces," Comm. Pure Appl. Math., vol. 34, iss. 6, pp. 741-797, 1981.
@article {schoen-simon, MRKEY = {0634285},
AUTHOR = {Schoen, Richard and Simon, Leon},
TITLE = {Regularity of stable minimal hypersurfaces},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {34},
YEAR = {1981},
NUMBER = {6},
PAGES = {741--797},
ISSN = {0010-3640},
CODEN = {CPAMAT},
MRCLASS = {49F22 (53C42 58E15)},
MRNUMBER = {0634285},
MRREVIEWER = {F. J. Almgren, Jr.},
DOI = {10.1002/cpa.3160340603},
ZBLNUMBER = {0497.49034},
} -
[shiohama-takagi]
K. Shiohama and R. Takagi, "A characterization of a standard torus in $E^{3}$," J. Differential Geometry, vol. 4, pp. 477-485, 1970.
@article {shiohama-takagi, MRKEY = {0276906},
AUTHOR = {Shiohama, Katsuhiro and Takagi, Ryoichi},
TITLE = {A characterization of a standard torus in {$E\sp{3}$}},
JOURNAL = {J. Differential Geometry},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {4},
YEAR = {1970},
PAGES = {477--485},
ISSN = {0022-040X},
MRCLASS = {53.75},
MRNUMBER = {0276906},
MRREVIEWER = {J. R. Vanstone},
URL = {http://projecteuclid.org/euclid.jdg/1214429643},
ZBLNUMBER = {0205.51402},
} -
[simon] L. Simon, Lectures on Geometric Measure Theory, Canberra: Australian National University Centre for Mathematical Analysis, 1983, vol. 3.
@book {simon, MRKEY = {0756417},
AUTHOR = {Simon, Leon},
TITLE = {Lectures on Geometric Measure Theory},
SERIES = {Proc. Centre Math. Anal., Australian National Univ.},
VOLUME = {3},
PUBLISHER = {Australian National University Centre for Mathematical Analysis},
ADDRESS = {Canberra},
YEAR = {1983},
PAGES = {vii+272},
ISBN = {0-86784-429-9},
MRCLASS = {49-01 (28A75 49F20)},
MRNUMBER = {0756417},
MRREVIEWER = {J. S. Joel},
ZBLNUMBER = {0546.49019},
} -
[simon93] L. Simon, "Existence of surfaces minimizing the Willmore functional," Comm. Anal. Geom., vol. 1, iss. 2, pp. 281-326, 1993.
@article {simon93, MRKEY = {1243525},
AUTHOR = {Simon, Leon},
TITLE = {Existence of surfaces minimizing the {W}illmore functional},
JOURNAL = {Comm. Anal. Geom.},
FJOURNAL = {Communications in Analysis and Geometry},
VOLUME = {1},
YEAR = {1993},
NUMBER = {2},
PAGES = {281--326},
ISSN = {1019-8385},
MRCLASS = {58E12 (49Q10 53A10)},
MRNUMBER = {1243525},
MRREVIEWER = {J. E. Brothers},
ZBLNUMBER = {0848.58012},
} -
[smyth] F. Smith, On the existence of embedded minimal 2Ðspheres in the 3–sphere, endowed with an arbitrary Riemannian metric, 1982.
@misc{smyth,
author={Smith, F.},
TITLE={On the existence of embedded minimal 2Ð-spheres in the 3--sphere, endowed with an arbitrary {R}iemannian metric},
NOTE={Ph.D. thesis, University of Melbourne, Supervisor: L. Simon},
YEAR={1982},
} -
[thomsen] G. Thomsen, "Über Konforme Geometrie, I: Grundlagen der Konformen Flächentheorie," Hamb. Math. Abh., vol. 3, pp. 31-56, 1923.
@article{thomsen,
author={Thomsen, G.},
TITLE={Über Konforme {G}eometrie, {I}: Grundlagen der Konformen {F}lächentheorie},
JOURNAL={Hamb. Math. Abh.},
VOLUME={3},
YEAR={1923},
PAGES={31--56},
JFMNUMBER = {49.0530.02},
} -
[topping]
P. Topping, "Towards the Willmore conjecture," Calc. Var. Partial Differential Equations, vol. 11, iss. 4, pp. 361-393, 2000.
@article {topping, MRKEY = {1808127},
AUTHOR = {Topping, Peter},
TITLE = {Towards the {W}illmore conjecture},
JOURNAL = {Calc. Var. Partial Differential Equations},
FJOURNAL = {Calculus of Variations and Partial Differential Equations},
VOLUME = {11},
YEAR = {2000},
NUMBER = {4},
PAGES = {361--393},
ISSN = {0944-2669},
MRCLASS = {53C65 (49Q05 53C42)},
MRNUMBER = {1808127},
MRREVIEWER = {Thomas I. Vogel},
DOI = {10.1007/s005260000042},
ZBLNUMBER = {1058.53060},
} -
[topping2005] P. Topping, "An approach to the Willmore conjecture," in Global Theory of Minimal Surfaces, Providence, RI: Amer. Math. Soc., 2005, vol. 2, pp. 769-772.
@incollection {topping2005, MRKEY = {2167288},
AUTHOR = {Topping, Peter},
TITLE = {An approach to the {W}illmore conjecture},
BOOKTITLE = {Global Theory of Minimal Surfaces},
SERIES = {Clay Math. Proc.},
VOLUME = {2},
PAGES = {769--772},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
MRCLASS = {53C42 (53A10)},
MRNUMBER = {2167288},
MRREVIEWER = {Bennett Palmer},
ZBLNUMBER = {1110.53004},
} -
[urbano]
F. Urbano, "Minimal surfaces with low index in the three-dimensional sphere," Proc. Amer. Math. Soc., vol. 108, iss. 4, pp. 989-992, 1990.
@article {urbano, MRKEY = {1007516},
AUTHOR = {Urbano, Francisco},
TITLE = {Minimal surfaces with low index in the three-dimensional sphere},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {108},
YEAR = {1990},
NUMBER = {4},
PAGES = {989--992},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {53C40},
MRNUMBER = {1007516},
MRREVIEWER = {Steen Markvorsen},
DOI = {10.2307/2047957},
ZBLNUMBER = {0691.53049},
} -
[weiner]
J. L. Weiner, "On a problem of Chen, Willmore, et al.," Indiana Univ. Math. J., vol. 27, iss. 1, pp. 19-35, 1978.
@article {weiner, MRKEY = {0467610},
AUTHOR = {Weiner, Joel L.},
TITLE = {On a problem of {C}hen, {W}illmore, et al.},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {27},
YEAR = {1978},
NUMBER = {1},
PAGES = {19--35},
ISSN = {0022-2518},
MRCLASS = {53C40 (58D10)},
MRNUMBER = {0467610},
MRREVIEWER = {T. J. Willmore},
DOI = {10.1512/iumj.1978.27.27003},
ZBLNUMBER = {0343.53038},
} -
[whitej]
J. H. White, "A global invariant of conformal mappings in space," Proc. Amer. Math. Soc., vol. 38, pp. 162-164, 1973.
@article {whitej, MRKEY = {0324603},
AUTHOR = {White, James H.},
TITLE = {A global invariant of conformal mappings in space},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {38},
YEAR = {1973},
PAGES = {162--164},
ISSN = {0002-9939},
MRCLASS = {53C45},
MRNUMBER = {0324603},
MRREVIEWER = {T. J. Willmore},
DOI = {10.2307/2038790},
ZBLNUMBER = {0256.53008},
} -
[willmore] T. J. Willmore, "Note on embedded surfaces," An. \c Sti. Univ. “Al. I. Cuza” Ia\c si Sec\c t. I a Mat., vol. 11B, pp. 493-496, 1965.
@article {willmore, MRKEY = {0202066},
AUTHOR = {Willmore, T. J.},
TITLE = {Note on embedded surfaces},
JOURNAL = {An. \c Sti. Univ. ``Al. I. Cuza'' Ia\c si Sec\c t. I a Mat. },
VOLUME = {11B},
YEAR = {1965},
PAGES = {493--496},
MRCLASS = {53.01 (53.74)},
MRNUMBER = {0202066},
MRREVIEWER = {H. B. Griffiths},
ZBLNUMBER = {0171.20001},
} -
[willmore71]
T. J. Willmore, "Mean curvature of Riemannian immersions," J. London Math. Soc., vol. 3, pp. 307-310, 1971.
@article {willmore71, MRKEY = {0283729},
AUTHOR = {Willmore, T. J.},
TITLE = {Mean curvature of {R}iemannian immersions},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {3},
YEAR = {1971},
PAGES = {307--310},
ISSN = {0024-6107},
MRCLASS = {53.74},
MRNUMBER = {0283729},
MRREVIEWER = {N. J. Hicks},
DOI = {10.1112/jlms/s2-3.2.307},
ZBLNUMBER = {0208.49802},
}