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Published online by Cambridge University Press:  13 May 2021

Áurea Casinhas Quintino
Affiliation:
Universidade Nova de Lisboa, Portugal
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Constrained Willmore Surfaces
Symmetries of a Möbius Invariant Integrable System
, pp. 240 - 244
Publisher: Cambridge University Press
Print publication year: 2021

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References

Bäcklund, A.V.: Concerning Surfaces with Constant Negative Curvature (translation; original: 1883). The New Era Printing Company, Lancaster, PA (1905).Google Scholar
Bianchi, L.: Lezioni di Geometria Differenziale. Enrico Spoerri Libraio-Editore, Pisa (1894).Google Scholar
Bianchi, L.: Ricerche sulle superficie isoterme e sulla deformazione delle quadriche. Annali di Matematica Pura ed Applicata 11, 93157 (1905).Google Scholar
Bianchi, L.: Complementi alle ricerche sulle superficie isoterme. Annali di Matematica Pura ed Applicata 12, 1954 (1906).CrossRefGoogle Scholar
Blaschke, W.J.E.: Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, band XXIX, Verlag von Julius Springer, Berlin (1929).Google Scholar
Bobenko, A.I.: Surfaces in terms of 2 by 2 matrices: old and new integrable cases. In: Fordy, A.P., Wood, J.C. (eds.) Harmonic Maps and Integrable Systems. Aspects of Mathematics, volume E23, 83–127. Vieweg Verlag, Braun-schweig/Wiesbaden (1994).Google Scholar
Bohle, C.: Möbius invariant flows in S4. Ph.D. thesis. Technische Universität Berlin (2003).Google Scholar
Bohle, C., Peters, G.P., Pinkall, U.: Constrained Willmore surfaces. Calculus of Variations and Partial Differential Equations 32(2), 263277 (2008).Google Scholar
Bonnet, P.O.: Mémoire sur la théorie des surfaces applicables sur une surface donnée. Journal de l’École Polytechnique 24, 209230 (1865).Google Scholar
Bryant, R.L.: A duality theorem for Willmore surfaces. Journal of Differential Geometry 20, 2353 (1984).Google Scholar
Burstall, F.E.: Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems. In: Terng, C.-L. (ed.) Integrable Systems, Geometry and Topology. AMS/IP Studies in Advanced Mathematics, volume 36, 1–82. American Mathematical Society, Providence, RI (2006).Google Scholar
Burstall, F.E., Calderbank, D.M.J.: Conformal submanifold geometry I–III. arXiv:1006.5700 [math.DG] (2010).Google Scholar
Burstall, F.E., Calderbank, D.M.J.: Conformal submanifold geometry IV–V. (In preparation).Google Scholar
Burstall, F.E., Donaldson, N.M., Pedit, F., Pinkall, U.: Isothermic submanifolds of symmetric R-spaces. Journal für die reine und angewandte Mathematik 660, 191243 (2011).Google Scholar
Burstall, F.E., Dorfmeister, J.F., Leschke, K., Quintino, A.: Darboux transforms and simple factor dressing of constant mean curvature surfaces. Manuscripta Mathematica 140, 213236 (2013).CrossRefGoogle Scholar
Burstall, F.E., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal Geometry of Surfaces in S4 and Quaternions. Lecture Notes in Mathematics, volume 1772. Springer-Verlag, Berlin (2002).CrossRefGoogle Scholar
Burstall, F.E., Pedit, F., Pinkall, U.: Schwarzian derivatives and flows of surfaces. In: Guest, M., Miyaoka, R., Ohnita, Y. (eds.) Differential Geometry and Integrable Systems. Contemporary Mathematics Series, volume 308, 39–61. American Mathematical Society, Providence, RI (2002).Google Scholar
Burstall, F.E., Quintino, A.C.: Dressing transformations of constrained Willmore surfaces. Communications in Analysis and Geometry 22, 469518 (2014).Google Scholar
Burstall, F.E., Rawnsley, J.H.: Twistor Theory for Riemannian Symmetric Spaces, with Applications to Harmonic Maps of Riemann Surfaces. Lecture Notes in Mathematics, volume 1424. Springer-Verlag, Berlin (1990).CrossRefGoogle Scholar
Burstall, F.E., Santos, S.D.: Special isothermic surfaces of type d. Journal of the London Mathematical Society 85, 571591 (2012).Google Scholar
Calapso, P.: Sulle superficie a linee di curvatura isoterme. Rendiconti Circolo Matematico di Palermo 17, 275286 (1903).Google Scholar
Calapso, P.: Sulle transformazioni delle superficie isoterme. Annali di Matematica Pura ed Applicata 24, 1148 (1915).Google Scholar
Chen, B.-Y.: Some conformal invariants of submanifolds and their applications. Bollettino dell’Unione Matematica Italiana 10(4), 380385 (1974).Google Scholar
Chern, S.-S.: Deformations of surfaces preserving principal curvatures. In: Chavel, I., Farkas, H.M. (eds.) Differential Geometry and Complex Analysis, Harry Ernest Rauch Memorial Volume, 155–163. Springer-Verlag, Berlin (1985).Google Scholar
Christoffel, E.B.: Über einige allgemeine Eigenshaften der Minimumsflächen. Crelle’s Journal 67, 218228 (1867).Google Scholar
Darboux, J.-G.: Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Gauthier-Villars, Paris (1887).Google Scholar
Darboux, J.-G.: Sur les surfaces isothermiques. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 128, 12991305 (1899).Google Scholar
Eells, J., Lemaire, L.: A report on harmonic maps. Bulletin of the London Mathematical Society 10, 168 (1978).Google Scholar
Eells, J., Lemaire, L.: Selected Topics in Harmonic Maps. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, number 50. American Mathematical Society, Providence, RI (1983).Google Scholar
Eells, J., Lemaire, L.: Another report on harmonic maps. Bulletin of the London Mathematical Society 20, 385524 (1988).CrossRefGoogle Scholar
Eells, J.B., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. American Journal of Mathematics 86(1), 109160 (1964).CrossRefGoogle Scholar
Ejiri, N.: Willmore surfaces with a duality in Sn (1). Proceedings of the London Mathematical Society (Series 3) 57(2), 383–416 (1988).Google Scholar
Euler, L.: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, sive, Solutio Problematis Isoperimetrici Latissimo Sensu Accepti. Marcum-Michaelem Bousquet & Socios, Lausannæ & Genevæ (1744).Google Scholar
Germain, M.-S.: Recherches sur la Théorie des Surfaces Élastiques. Mme Ve Courcier, Libraire pour les Sciences, Paris (1821).Google Scholar
Germain, M.-S.: Remarques sur la Nature, les Bornes et l’Étendue de la Question des Surfaces Élastiques, et Équation Générale de ces Surfaces. Imprimerie de Huzard-Courcier, Paris (1826).Google Scholar
Germain, M.-S.: Mémoire sur la courbure des surfaces. Crelle’s Journal 7, 129 (1831).Google Scholar
Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames. Cambridge Tracts in Mathematics, volume 150. Cambridge University Press, Cambridge (2002).Google Scholar
Hertrich-Jeromin, U.: Introduction to Möbius Differential Geometry. London Mathematical Society Lecture Note Series, volume 300. Cambridge University Press, Cambridge (2003).CrossRefGoogle Scholar
Hertrich-Jeromin, U., Pedit, F.: Remarks on Darboux transforms of isothermic surfaces. Documenta Mathematica 2, 313333 (1997).CrossRefGoogle Scholar
Inoguchi, J.-I., Kobayashi, S.: Characterizations of Bianchi–Bäcklund transformations of constant mean curvature surfaces. International Journal of Mathematics 16(2), 101110 (2005).Google Scholar
Jost, J.: Compact Riemann Surfaces: An Introduction to Contemporary Mathematics. Universitext. Springer-Verlag, Berlin (2006).Google Scholar
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, volume 1. Wiley Classics Library. Wiley-Interscience, John Wiley & Sons, Inc., New York (1996).Google Scholar
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, volume 2. Wiley Classics Library. Wiley-Interscience, John Wiley & Sons, Inc., New York (1996).Google Scholar
Lagrange, J.-L.: Essai d’une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies. Mélanges de Philosophie et de Mathématique de la Société Royale de Turin (Miscellanea Taurinensia), 2, 335–362 (1762).Google Scholar
Landau, L., Lifshitz, E.: Lehrbuch der Theoretischen Physic, band VII, Elastizitätstheorie. Akademie-Verlag, Berlin (1965).Google Scholar
Langer, J., Singer, D.A.: Curves in the hyperbolic plane and mean curvature of tori in 3-space. Bulletin of the London Mathematical Society 16, 531534 (1984).CrossRefGoogle Scholar
Lawson, B.: Complete minimal surfaces in S3. Annals of Mathematics 92(3), 335–74 (1970).CrossRefGoogle Scholar
Lipowsky, R.: Kooperatives verhalten von membranen. Physics Bulletin 52, 555560 (1996).Google Scholar
Machado, A.: Geometria Diferencial: Uma Introdução Fundamental. Textos de Matemática, volume 9. Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa (1991).Google Scholar
Machado, A.: Tópicos de Análise e Topologia em Variedades. Textos de Matemática, volume 8. Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa (1991).Google Scholar
Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Annals of Mathematics 179(2), 683782 (2014).CrossRefGoogle Scholar
Meusnier, J.B.M.C.: Mémoire sur la courbure des surfaces. Mémoires de Mathématique et de Physique, présentés à l’Académie Royale des Sciences de Paris, par divers Savans, & lûs dans ses Assemblées 10, 477510 (1785).Google Scholar
Narasimhan, R.: Compact Riemann Surfaces. Lectures in Mathematics. ETH Zürich, Birkhäuser, Basel (1992).Google Scholar
O’Neill, B..: Semi-Riemannian Geometry with Applications to Relativity. Pure and Applied Mathematics, volume 103. Academic Press, San Diego, CA (1983).Google Scholar
O’Neill, B.: Elementary Differential Geometry (second edition). Academic Press, San Diego, CA (1997).Google Scholar
Noether, E.: Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 235–257 (1918).Google Scholar
Palmer, B.: Isothermic surfaces and the Gauss map. Proceedings of the American Mathematical Society 104, 876–884 (1988).Google Scholar
[58] Pinkall, U.: Hopf tori in S3. Inventiones Mathematicae 81, 379386 (1985).Google Scholar
Plateau, J.A.F.: Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires. Gauthier-Villars, Paris (1873).Google Scholar
Poisson, S.D.: Mémoire sur les surfaces élastiques. Mémoires de la Classe des Sciences Mathématiques et Physiques de l’Institut National de France, Année 1812, Seconde Partie, 167–225 (1812).Google Scholar
Quintino, A.C.: Transformations of generalized harmonic bundles and constrained Willmore surfaces. In: Toda, M.D. (ed.) Willmore Energy and Willmore Conjecture, 9–47. Chapman & Hall/CRC Monographs and Research Notes in Mathematics. CRC Press, Taylor and Francis Group, Boca Raton, FL (2017).Google Scholar
Quintino, A.C., Santos, S.D.: Polynomial conserved quantities for constrained Willmore surfaces. arXiv:1507.01253v1 [mathDG] (2015).Google Scholar
Richter, J.: Conformal maps of a Riemann surface into the space of quaternions. Ph.D. thesis, Technische Universität Berlin (1997).Google Scholar
Rigoli, M.: The conformal Gauss map of submanifolds of the Möbius space. Annals of Global Analysis and Geometry 5(2), 97116 (1987).Google Scholar
Santos, S.D.: Special isothermic surfaces. Ph.D. thesis, University of Bath (2008).Google Scholar
Sterling, I., Wente, H.: Existence and classification of constant mean curvature multibubbletons of finite and infinite type. Indiana University Mathematics Journal 42(4), 12391266 (1993).Google Scholar
Terng, C.-L., Uhlenbeck, K.: Bäcklund transformations and loop group actions. Communications on Pure and Applied Mathematics 53, 175 (2000).Google Scholar
Thomsen, G.: Über konforme Geometrie I: Grundlagen der konformen Flächentheorie. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 3, 3156 (1923).Google Scholar
Uhlenbeck, K.: Harmonic maps into Lie groups: classical solutions of the chiral model. Journal of Differential Geometry 30, 150 (1989).Google Scholar
Weiner, J.L.: On a problem of Chen, Willmore, et al.. Indiana University Mathematics Journal 27(1), 1935 (1978).Google Scholar
White, J.H.: A global invariant of conformal mappings in space. Proceedings of the American Mathematical Society 38, 162164 (1973).Google Scholar
Willmore, T.J.: Note on embedded surfaces. Analele Ştiinţifice ale Universitaţii “Alexandru Ioan Cuza” din Iaşi (Serie Noua), Sect. I a Mat. 11B, 493–496 (1965).Google Scholar
Willmore, T.J.: Riemannian Geometry. Oxford Science Publications. Clarendon Press/Oxford University Press, New York (1993).CrossRefGoogle Scholar

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  • References
  • Áurea Casinhas Quintino, Universidade Nova de Lisboa, Portugal
  • Book: Constrained Willmore Surfaces
  • Online publication: 13 May 2021
  • Chapter DOI: https://doi.org/10.1017/9781108885478.014
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  • References
  • Áurea Casinhas Quintino, Universidade Nova de Lisboa, Portugal
  • Book: Constrained Willmore Surfaces
  • Online publication: 13 May 2021
  • Chapter DOI: https://doi.org/10.1017/9781108885478.014
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Áurea Casinhas Quintino, Universidade Nova de Lisboa, Portugal
  • Book: Constrained Willmore Surfaces
  • Online publication: 13 May 2021
  • Chapter DOI: https://doi.org/10.1017/9781108885478.014
Available formats
×