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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A variational problem for surfaces in
Laguerre geometry

Authors: Emilio Musso and Lorenzo Nicolodi
Journal: Trans. Amer. Math. Soc. 348 (1996), 4321-4337
MSC (1991): Primary 58E40, 53A40, 53A05
MathSciNet review: 1370648
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Abstract: We consider the variational problem defined by the functional $\int {\frac {{H^{2}-K}}{{K}}}dA$ on immersed surfaces in Euclidean space. Using the invariance of the functional under the group of Laguerre transformations, we study the extremal surfaces by the method of moving frames.

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  • 1. Luigi Bianchi, Opere. Vol. 11. Corrispondenza, A cura dell’Unione Matematica Italiana e col contributo dell Consiglio Nazionale delle Ricerche, Edizioni Cremonese, Rome, 1959 (Italian). MR 0130160
  • 2. W. Blaschke, Über die Geometrie von Laguerre: I. Grundformeln der Flächentheorie, Abh. Math. Sem. Univ. Hamburg 3 (1924), 176--194.
  • 3. ------, Über die Geometrie von Laguerre: II. Flächentheorie in Ebenenkoordinaten, Abh. Math. Sem. Univ. Hamburg 3 (1924), 195--212.
  • 4. ------, Über die Geometrie von Laguerre: III. Beiträge zur Flächentheorie, Abh. Math. Sem. Univ. Hamburg 4 (1925), 1--12.
  • 5. ------, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, B. 3, bearbeitet von G. Thomsen, J. Springer, Berlin, 1929.
  • 6. Robert L. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), no. 1, 23–53. MR 772125
  • 7. R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. MR 1083148
  • 8. E. Cartan, Théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Gauthier-Villars, Paris, 1937.
  • 9. Thomas E. Cecil, Lie sphere geometry, Universitext, Springer-Verlag, New York, 1992. With applications to submanifolds. MR 1219311
  • 10. P. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775–814. MR 0410607
  • 11. Gary R. Jensen, Deformation of submanifolds of homogeneous spaces, J. Differential Geom. 16 (1981), no. 2, 213–246. MR 638789
  • 12. Tilla Klotz Milnor, Harmonic maps and classical surface theory in Minkowski 3-space, Trans. Amer. Math. Soc. 280 (1983), no. 1, 161–185. MR 712254,
  • 13. E. Musso, L. Nicolodi, $L$-minimal canal surfaces, Rend. Matematica 15 (1995), 421--445. CMP 96:04
  • 14. E. Musso, L. Nicolodi, Isothermal surfaces in Laguerre geometry, Boll. Un. Mat. Ital. (to appear).

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Additional Information

Emilio Musso
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università di L’Aquila, via Vetoio, I-67010 Coppito, L’ Aquila, Italy

Lorenzo Nicolodi
Affiliation: Dipartimento di Matematica “G. Castelnuovo", Università di Roma “La Sapienza", p.le A. Moro 2, I-00185 Roma, Italy

Keywords: Laguerre geometry, $L$-minimal surfaces, Legendre surfaces
Received by editor(s): June 16, 1994
Additional Notes: Partially supported by CNR contract n. 93.00554.CTO1, the GADGET initiative of the EC and MURST 40%.
Article copyright: © Copyright 1996 American Mathematical Society