Abstract
We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into ${\mathbb R}^m$ with uniformly bounded energy and nondegenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the Möbius group action, below some energy threshold.
Note: To view the article, click on the URL link for the DOI number.
-
[Aub]
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, New York: Springer-Verlag, 1982, vol. 252.@book {Aub, MRKEY = {0681859},
AUTHOR = {Aubin, Thierry},
TITLE = {Nonlinear Analysis on Manifolds. {M}onge-{A}mpère Equations},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {252},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1982},
PAGES = {xii+204},
ISBN = {0-387-90704-1},
MRCLASS = {58-02 (35J60 35N99 53-02 58G30)},
MRNUMBER = {0681859},
MRREVIEWER = {A. G. Ramm},
ZBLNUMBER = {0512.53044},
DOI = {10.1007/978-1-4612-5734-9},
} -
[BK] 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 {BK, 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},
ZBLNUMBER = {1029.53073},
DOI = {10.1155/S1073792803208072},
VOLUME = {2003},
} -
[B] Y. Bernard, Noether’s Theorem and the Willmore functional, 2013.
@misc{B,
author={Bernard, Yann},
TITLE={Noether's Theorem and the {W}illmore functional},
NOTE={preprint},
YEAR={2013},
} -
[BR1] Y. Bernard and T. Rivière, "Local Palais-Smale sequences for the Willmore functional," Comm. Anal. Geom., vol. 19, iss. 3, pp. 563-599, 2011.
@article {BR1, MRKEY = {2843242},
AUTHOR = {Bernard, Yann and Rivi{è}re, Tristan},
TITLE = {Local {P}alais-{S}male sequences for the {W}illmore functional},
JOURNAL = {Comm. Anal. Geom.},
FJOURNAL = {Communications in Analysis and Geometry},
VOLUME = {19},
YEAR = {2011},
NUMBER = {3},
PAGES = {563--599},
ISSN = {1019-8385},
MRCLASS = {58E12 (35J47 35J50 35J60 53C42)},
MRNUMBER = {2843242},
MRREVIEWER = {Andreas Gastel},
ZBLNUMBER = {1245.53051},
DOI = {10.4310/CAG.2011.v19.n3.a5},
} -
[BR2] Y. Bernard and T. Rivière, "Singularity removability at branch points for Willmore surfaces," Pacific J. Math., vol. 265, iss. 2, pp. 257-311, 2013.
@article {BR2, MRKEY = {3096502},
AUTHOR = {Bernard, Yann and Rivi{è}re, Tristan},
TITLE = {Singularity removability at branch points for {W}illmore surfaces},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {265},
YEAR = {2013},
NUMBER = {2},
PAGES = {257--311},
ISSN = {0030-8730},
MRCLASS = {58E30 (35B44 35B60 35J47 35R01 53Axx 58E15 58K99)},
MRNUMBER = {3096502},
DOI = {10.2140/pjm.2013.265.257},
ZBLNUMBER = {06242169},
} -
[Bla] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische grundlagen von Einsteins relativitätstheorie, III," in Differentialgeometrie der Kreise und Kugeln, Gerhard Thomsen, 1929, vol. 39.
@incollection{Bla,
author={Blaschke, Wilhelm},
TITLE={Vorlesungen über {D}ifferentialgeometrie und geometrische grundlagen von {E}insteins relativitätstheorie, {III}},
BOOKTITLE={Differentialgeometrie der {K}reise und {K}ugeln},
SERIES={Grundlehren Math. Wissen.},
VOLUME={39},
PUBLISHER={Gerhard Thomsen},
YEAR={1929},
JFMNUMBER = {55.0422.01},
} -
[Bry] R. L. Bryant, "A duality theorem for Willmore surfaces," J. Differential Geom., vol. 20, iss. 1, pp. 23-53, 1984.
@article {Bry, 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},
ZBLNUMBER = {0555.53002},
URL = {http://projecteuclid.org/euclid.jdg/1214438991},
} -
[BuPa] A. Butscher and F. Pacard, "Doubling constant mean curvature tori in $S^3$," Ann. Sc. Norm. Super. Pisa Cl. Sci., vol. 5, iss. 4, pp. 611-638, 2006.
@article {BuPa, MRKEY = {2297724},
AUTHOR = {Butscher, Adrian and Pacard, Frank},
TITLE = {Doubling constant mean curvature tori in {$S\sp 3$}},
JOURNAL = {Ann. Sc. Norm. Super. Pisa Cl. Sci.},
FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V},
VOLUME = {5},
YEAR = {2006},
NUMBER = {4},
PAGES = {611--638},
ISSN = {0391-173X},
MRCLASS = {53A10},
MRNUMBER = {2297724},
MRREVIEWER = {Jesse Ratzkin},
ZBLNUMBER = {1170.53303},
URL = {http://www.numdam.org/item?id=ASNSP_2006_5_5_4_611_0},
} -
[DiT] W. Ding and G. Tian, "Energy identity for a class of approximate harmonic maps from surfaces," Comm. Anal. Geom., vol. 3, iss. 3-4, pp. 543-554, 1995.
@article {DiT, MRKEY = {1371209},
AUTHOR = {Ding, Weiyue and Tian, Gang},
TITLE = {Energy identity for a class of approximate harmonic maps from surfaces},
JOURNAL = {Comm. Anal. Geom.},
FJOURNAL = {Communications in Analysis and Geometry},
VOLUME = {3},
YEAR = {1995},
NUMBER = {3-4},
PAGES = {543--554},
ISSN = {1019-8385},
MRCLASS = {58E20 (58G11)},
MRNUMBER = {1371209},
MRREVIEWER = {Joseph F. Grotowski},
ZBLNUMBER = {0855.58016},
} -
[Ge] Y. Ge, "A remark on generalized harmonic maps into spheres," Nonlinear Anal., vol. 36, iss. 4, Ser. A: Theory Methods, pp. 495-506, 1999.
@article {Ge, MRKEY = {1675268},
AUTHOR = {Ge, Yuxin},
TITLE = {A remark on generalized harmonic maps into spheres},
JOURNAL = {Nonlinear Anal.},
FJOURNAL = {Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal. Series A: Theory and Methods},
VOLUME = {36},
YEAR = {1999},
NUMBER = {4, Ser. A: Theory Methods},
PAGES = {495--506},
ISSN = {0362-546X},
CODEN = {NOANDD},
MRCLASS = {58E20 (35J60 49N60)},
MRNUMBER = {1675268},
MRREVIEWER = {Naoyuki Ishimura},
ZBLNUMBER = {0932.35071},
DOI = {10.1016/S0362-546X(98)00079-0},
} -
[Hel] F. Hélein, Harmonic Maps, Conservation Laws and Moving Frames, Second ed., Cambridge: Cambridge Univ. Press, 2002, vol. 150.
@book {Hel, MRKEY = {1913803},
AUTHOR = {H{é}lein, Fr{é}d{é}ric},
TITLE = {Harmonic Maps, Conservation Laws and Moving Frames},
SERIES = {Cambridge Tracts in Math.},
VOLUME = {150},
EDITION = {Second},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2002},
PAGES = {xxvi+264},
ISBN = {0-521-81160-0},
MRCLASS = {58E20 (35A22 35J15 53C43 58E12)},
MRNUMBER = {1913803},
MRREVIEWER = {Andreas Gastel},
ZBLNUMBER = {1010.58010},
DOI = {10.1017/CBO9780511543036},
} -
[Jo] J. Jost, Two-Dimensional Geometric Variational Problems, Chichester: John Wiley & Sons Ltd., 1991.
@book {Jo, MRKEY = {1100926},
AUTHOR = {Jost, J{ü}rgen},
TITLE = {Two-Dimensional Geometric Variational Problems},
SERIES = {Pure Appl. Math. (NY)},
NOTE = {A Wiley-Interscience Publication},
PUBLISHER = {John Wiley \& Sons Ltd.},
ADDRESS = {Chichester},
YEAR = {1991},
PAGES = {x+236},
ISBN = {0-471-92839-9},
MRCLASS = {58E12 (32G15 49Q05 58-02 58E20)},
MRNUMBER = {1100926},
MRREVIEWER = {Wei Yue Ding},
ZBLNUMBER = {0729.49001},
} -
[KuLi] E. Kuwert and Y. Li, "$W^{2,2}$-conformal immersions of a closed Riemann surface into $\Bbb R^n$," Comm. Anal. Geom., vol. 20, iss. 2, pp. 313-340, 2012.
@article {KuLi, MRKEY = {2928715},
AUTHOR = {Kuwert, Ernst and Li, Yuxiang},
TITLE = {{$W\sp {2,2}$}-conformal immersions of a closed {R}iemann surface into {$\Bbb R\sp n$}},
JOURNAL = {Comm. Anal. Geom.},
FJOURNAL = {Communications in Analysis and Geometry},
VOLUME = {20},
YEAR = {2012},
NUMBER = {2},
PAGES = {313--340},
ISSN = {1019-8385},
MRCLASS = {53A30 (53A07)},
MRNUMBER = {2928715},
MRREVIEWER = {Peng Wang},
ZBLNUMBER = {1271.53010},
DOI = {10.4310/CAG.2012.v20.n2.a4},
} -
[KS] E. Kuwert and R. Schätzle, "Removability of point singularities of Willmore surfaces," Ann. of Math., vol. 160, iss. 1, pp. 315-357, 2004.
@article {KS, 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},
ZBLNUMBER = {1078.53007},
DOI = {10.4007/annals.2004.160.315},
} -
[KS1] E. Kuwert and R. Schätzle, "Closed surfaces with bounds on their Willmore energy," Ann. Sc. Norm. Super. Pisa Cl. Sci., vol. 11, iss. 3, pp. 605-634, 2012.
@article {KS1, MRKEY = {3059839},
AUTHOR = {Kuwert, Ernst and Sch{ä}tzle, Reiner},
TITLE = {Closed surfaces with bounds on their {W}illmore energy},
JOURNAL = {Ann. Sc. Norm. Super. Pisa Cl. Sci.},
FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V},
VOLUME = {11},
YEAR = {2012},
NUMBER = {3},
PAGES = {605--634},
ISSN = {0391-173X},
MRCLASS = {53A30 (49Q15 53C21)},
MRNUMBER = {3059839},
ZBLNUMBER = {1260.53027},
URL = {http://annaliscienze.sns.it/index.php?page=Article&id=251},
} -
[KS2] E. Kuwert and R. Schätzle, "Minimizers of the Willmore functional under fixed conformal class," J. Differential Geom., vol. 93, iss. 3, pp. 471-530, 2013.
@article {KS2, MRKEY = {3024303},
AUTHOR = {Kuwert, Ernst and Sch{ä}tzle, Reiner},
TITLE = {Minimizers of the {W}illmore functional under fixed conformal class},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {93},
YEAR = {2013},
NUMBER = {3},
PAGES = {471--530},
ISSN = {0022-040X},
MRCLASS = {53C42 (53A30 58E30)},
MRNUMBER = {3024303},
MRREVIEWER = {Marian-Ioan Munteanu},
ZBLNUMBER = {06201325},
URL = {http://projecteuclid.org/euclid.jdg/1361844942},
} -
[LaRi] P. Laurain and T. Rivière, Angular Energy Quantization for Linear Elliptic Systems with Antisymmetric Potentials and Applications.
@misc{LaRi,
author={Laurain, Paul and Rivière, Tristan},
TITLE={Angular Energy Quantization for Linear Elliptic Systems with Antisymmetric Potentials and Applications},
NOTE={to appear in \emph{Anal. PDE}},
ARXIV={1109.3599},
} -
[LiRi1] F. Lin and T. Rivière, "A quantization property for moving line vortices," Comm. Pure Appl. Math., vol. 54, iss. 7, pp. 826-850, 2001.
@article {LiRi1, MRKEY = {1823421},
AUTHOR = {Lin, Fang-Hua and Rivi{è}re, Tristan},
TITLE = {A quantization property for moving line vortices},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {54},
YEAR = {2001},
NUMBER = {7},
PAGES = {826--850},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {35K55 (35B25 49J10)},
MRNUMBER = {1823421},
MRREVIEWER = {Yisong Yang},
ZBLNUMBER = {1029.35127},
DOI = {10.1002/cpa.3003},
} -
[LiRi2] F. Lin and T. Rivière, "Energy quantization for harmonic maps," Duke Math. J., vol. 111, iss. 1, pp. 177-193, 2002.
@article {LiRi2, MRKEY = {1876445},
AUTHOR = {Lin, Fang-Hua and Rivi{è}re, Tristan},
TITLE = {Energy quantization for harmonic maps},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {111},
YEAR = {2002},
NUMBER = {1},
PAGES = {177--193},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {58E20},
MRNUMBER = {1876445},
MRREVIEWER = {Ernst C. Kuwert},
ZBLNUMBER = {1014.58008},
DOI = {10.1215/S0012-7094-02-11116-8},
} -
[MN] F. C. Marques and A. Neves, "Min-max theory and the Willmore conjecture," Ann. of Math., vol. 179, pp. 683-782, 2014.
@article{MN,
author={Marques, F. C. and Neves, A.},
TITLE={Min-max theory and the {W}illmore conjecture},
JOURNAL={Ann. of Math.},
VOLUME={179},
YEAR={2014},
DOI={10.4007/annals.2014.179.2.6},
PAGES={683--782},
} -
[Mon] S. Montiel, "Willmore two-spheres in the four-sphere," Trans. Amer. Math. Soc., vol. 352, iss. 10, pp. 4469-4486, 2000.
@article {Mon, MRKEY = {1695032},
AUTHOR = {Montiel, Sebasti{á}n},
TITLE = {Willmore two-spheres in the four-sphere},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {352},
YEAR = {2000},
NUMBER = {10},
PAGES = {4469--4486},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {53C42 (53C28)},
MRNUMBER = {1695032},
MRREVIEWER = {Udo Hertrich-Jeromin},
ZBLNUMBER = {0961.53035},
DOI = {10.1090/S0002-9947-00-02571-X},
} -
[MS] S. Müller and V. vSverák, "On surfaces of finite total curvature," J. Differential Geom., vol. 42, iss. 2, pp. 229-258, 1995.
@article {MS, MRKEY = {1366547},
AUTHOR = {M{ü}ller, S. and {Š}ver{á}k, V.},
TITLE = {On surfaces of finite total curvature},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {42},
YEAR = {1995},
NUMBER = {2},
PAGES = {229--258},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53A05 (53C21)},
MRNUMBER = {1366547},
MRREVIEWER = {Tatiana Toro},
ZBLNUMBER = {0853.53003},
URL = {http://projecteuclid.org/euclid.jdg/1214457233},
} -
[Pa] T. H. Parker, "Bubble tree convergence for harmonic maps," J. Differential Geom., vol. 44, iss. 3, pp. 595-633, 1996.
@article {Pa, MRKEY = {1431008},
AUTHOR = {Parker, Thomas H.},
TITLE = {Bubble tree convergence for harmonic maps},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {44},
YEAR = {1996},
NUMBER = {3},
PAGES = {595--633},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {58E20},
MRNUMBER = {1431008},
MRREVIEWER = {Daniel Pollack},
ZBLNUMBER = {0874.58012},
URL = {http://projecteuclid.org/euclid.jdg/1214459224},
} -
[Ri1] T. Rivière, Weak Immersions of surfaces with ${L}^2$ bounded second fundamental form, 2013.
@misc{Ri1,
author = {Rivi{è}re, Tristan},
TITLE = {Weak Immersions of surfaces with ${L}^2$ bounded second fundamental form},
NOTE={{IAS/P}ark {C}ity {M}athematics {I}nstitute summer school},
YEAR={2013},
} -
[Ri2] T. Rivière, "Analysis aspects of Willmore surfaces," Invent. Math., vol. 174, iss. 1, pp. 1-45, 2008.
@article {Ri2, 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},
ZBLNUMBER = {1155.53031},
DOI = {10.1007/s00222-008-0129-7},
} -
[Ri3] T. Rivière, Variational Principles for immersed Surfaces with ${L}^2$-bounded Second Fundamental Form.
@misc{Ri3, KEY={Riv13c},
AUTHOR = {Rivi{è}re, Tristan},
TITLE = {Variational Principles for immersed Surfaces with ${L}^2$-bounded Second Fundamental Form},
NOTE={\emph{J. reine angew. Math.},
published online: 01/23/2013},
SORTYEAR={2014},
DOI = {10.1515/crelle-2012-0106},
} -
[Ri4] T. Rivière, "Lipschitz conformal immersions from degenerating Riemann surfaces with $L^2$-bounded second fundamental forms," Adv. Calc. Var., vol. 6, iss. 1, pp. 1-31, 2013.
@article {Ri4, MRKEY = {3008339},
AUTHOR = {Rivi{è}re, Tristan},
TITLE = {Lipschitz conformal immersions from degenerating {R}iemann surfaces with {$L\sp 2$}-bounded second fundamental forms},
JOURNAL = {Adv. Calc. Var.},
FJOURNAL = {Advances in Calculus of Variations},
VOLUME = {6},
YEAR = {2013},
NUMBER = {1},
PAGES = {1--31},
ISSN = {1864-8258},
MRCLASS = {58E30 (35J35 35J48 35J50 49Q10 53A30)},
MRNUMBER = {3008339},
MRREVIEWER = {Rossella Bartolo},
ZBLNUMBER = {06137860},
DOI = {10.1515/acv-2012-0108},
} -
[Ri5] T. Rivière, "Interpolation spaces and energy quantization for Yang-Mills fields," Comm. Anal. Geom., vol. 10, iss. 4, pp. 683-708, 2002.
@article {Ri5, MRKEY = {1925499},
AUTHOR = {Rivi{è}re, Tristan},
TITLE = {Interpolation spaces and energy quantization for {Y}ang-{M}ills fields},
JOURNAL = {Comm. Anal. Geom.},
FJOURNAL = {Communications in Analysis and Geometry},
VOLUME = {10},
YEAR = {2002},
NUMBER = {4},
PAGES = {683--708},
ISSN = {1019-8385},
MRCLASS = {58E15 (81T13)},
MRNUMBER = {1925499},
MRREVIEWER = {J{ü}rgen Eichhorn},
ZBLNUMBER = {1018.58006},
} -
[Ri6] T. Rivière, "Bubbling and regularity issues in geometric non-linear analysis," in Proceedings of the International Congress of Mathematicians, Vol. III, Beijing, 2002, pp. 197-208.
@inproceedings {Ri6, MRKEY = {1957532},
AUTHOR = {Rivi{è}re, Tristan},
TITLE = {Bubbling and regularity issues in geometric non-linear analysis},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {III}},
VENUE={{B}eijing, 2002},
PAGES = {197--208},
PUBLISHER = {Higher Ed. Press},
ADDRESS = {Beijing},
YEAR = {2002},
MRCLASS = {35J60 (35J20 49Q20 58E15 58E20)},
MRNUMBER = {1957532},
MRREVIEWER = {Jie Yang},
ZBLNUMBER = {1136.35334},
} -
[SaU] J. Sacks and K. Uhlenbeck, "The existence of minimal immersions of $2$-spheres," Ann. of Math., vol. 113, iss. 1, pp. 1-24, 1981.
@article {SaU, MRKEY = {0604040},
AUTHOR = {Sacks, J. and Uhlenbeck, K.},
TITLE = {The existence of minimal immersions of {$2$}-spheres},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {113},
YEAR = {1981},
NUMBER = {1},
PAGES = {1--24},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {58E12 (53C42 58E20)},
MRNUMBER = {0604040},
MRREVIEWER = {John C. Wood},
ZBLNUMBER = {0462.58014},
DOI = {10.2307/1971131},
} -
[Sch] R. M. Schätzle, "Estimation of the conformal factor under bounded Willmore energy," Math. Z., vol. 274, iss. 3-4, pp. 1341-1383, 2013.
@article {Sch, MRKEY = {3078270},
AUTHOR = {Sch{ä}tzle, Reiner Michael},
TITLE = {Estimation of the conformal factor under bounded {W}illmore energy},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {274},
YEAR = {2013},
NUMBER = {3-4},
PAGES = {1341--1383},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {53A05 (53A30 53C21)},
MRNUMBER = {3078270},
ZBLNUMBER = {1275.53011},
DOI = {10.1007/s00209-012-1119-4},
} -
[ScU] R. Schoen and K. Uhlenbeck, "Boundary regularity and the Dirichlet problem for harmonic maps," J. Differential Geom., vol. 18, iss. 2, pp. 253-268, 1983.
@article {ScU, MRKEY = {0710054},
AUTHOR = {Schoen, Richard and Uhlenbeck, Karen},
TITLE = {Boundary regularity and the {D}irichlet problem for harmonic maps},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {18},
YEAR = {1983},
NUMBER = {2},
PAGES = {253--268},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {58E20},
MRNUMBER = {0710054},
MRREVIEWER = {Liane Valere Bouche},
ZBLNUMBER = {0547.58020},
URL = {http://projecteuclid.org/euclid.jdg/1214437663},
} -
[Si] L. Simon, "Existence of surfaces minimizing the Willmore functional," Comm. Anal. Geom., vol. 1, iss. 2, pp. 281-326, 1993.
@article {Si, 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},
} -
[St] M. Struwe, "On the evolution of harmonic mappings of Riemannian surfaces," Comment. Math. Helv., vol. 60, iss. 4, pp. 558-581, 1985.
@article {St, MRKEY = {0826871},
AUTHOR = {Struwe, Michael},
TITLE = {On the evolution of harmonic mappings of {R}iemannian surfaces},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {60},
YEAR = {1985},
NUMBER = {4},
PAGES = {558--581},
ISSN = {0010-2571},
CODEN = {COMHAX},
MRCLASS = {58E20 (53C42)},
MRNUMBER = {0826871},
MRREVIEWER = {S. M. Salamon},
ZBLNUMBER = {0595.58013},
DOI = {10.1007/BF02567432},
} -
[To] T. Toro, "Geometric conditions and existence of bi-Lipschitz parameterizations," Duke Math. J., vol. 77, iss. 1, pp. 193-227, 1995.
@article {To, MRKEY = {1317632},
AUTHOR = {Toro, Tatiana},
TITLE = {Geometric conditions and existence of bi-{L}ipschitz parameterizations},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {77},
YEAR = {1995},
NUMBER = {1},
PAGES = {193--227},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {28A78},
MRNUMBER = {1317632},
MRREVIEWER = {J. G. Krzy{\.z}},
ZBLNUMBER = {0847.42011},
DOI = {10.1215/S0012-7094-95-07708-4},
} -
[Zhu] M. Zhu, "Harmonic maps from degenerating Riemann surfaces," Math. Z., vol. 264, iss. 1, pp. 63-85, 2010.
@article {Zhu, MRKEY = {2564932},
AUTHOR = {Zhu, Miaomiao},
TITLE = {Harmonic maps from degenerating {R}iemann surfaces},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {264},
YEAR = {2010},
NUMBER = {1},
PAGES = {63--85},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {58E20},
MRNUMBER = {2564932},
MRREVIEWER = {Andreas Gastel},
ZBLNUMBER = {1213.53086},
DOI = {10.1007/s00209-008-0452-0},
}
