Advertisement

Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 1089–1110 | Cite as

Holomorphic differentials and Laguerre deformation of surfaces

  • Emilio Musso
  • Lorenzo NicolodiEmail author
Article

Abstract

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their L-Gauss maps, the T-transforms of L-minimal isothermic surfaces have constant mean curvature \(H=r\) in some translate of hyperbolic 3-space \({\mathbb {H}}^3(-r^2)\subset \mathbb {R}^4_1\), de Sitter 3-space \({\mathbb {S}}^3_1(r^2)\subset \mathbb {R}^4_1\), or have mean curvature \(H=0\) in some translate of a time-oriented lightcone in \(\mathbb {R}^4_1\). As an application, we show that various instances of the Lawson isometric correspondence can be viewed as special cases of the T-transformation of L-isothermic surfaces with holomorphic quartic differential.

Keywords

Laguerre surface geometry Laguerre minimal surfaces Laguerre Gauss map L-isothermic surfaces Constant mean curvature surfaces Minimal surfaces Maximal surfaces Isotropic geometry Lawson correspondence 

Mathematics Subject Classification

53A35 53C42 

References

  1. 1.
    Aiyama, R., Akutagawa, K.: Kenmotsu–Bryant type representation formulas for constant mean curvature surfaces in \(H^3(-c^2)\) and \(S^3_1(c^2)\). Ann. Global Anal. Geom. 17(1), 49–75 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aledo, J.A., Gálvez, J.A., Mira, P.: Marginally trapped surfaces in \(\mathbb{L}^4\) and an extended Weierstrass–Bryant representation. Ann. Global Anal. Geom. 28(4), 395–415 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bianchi, L.: Complementi alle ricerche sulle superficie isoterme. Ann. Mat. Pura Appl. 12, 19–54 (1905)zbMATHCrossRefGoogle Scholar
  4. 4.
    Blaschke, W.: Über die Geometrie von Laguerre: I. Abh. Math. Sem. Univ. Hamburg 3, 176–194 (1924)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Blaschke, W.: Über die Geometrie von Laguerre: II. Abh. Math. Sem. Univ. Hamburg 3, 195–212 (1924)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blaschke, W.: Über die Geometrie von Laguerre: III. Abh. Math. Sem. Univ. Hamburg 4, 1–12 (1925)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Blaschke, G.: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie III. Differentialgeometrie der Kreise und Kugeln. In: Thomsen, G. (ed.) Die Grundlehren der Mathematischen Wissenschaften, vol. 29. Springer, Berlin (1929)Google Scholar
  8. 8.
    Bobenko, A.I., Hoffmann, T., Springborn, B.A.: Minimal surfaces from circle patterns: geometry from combinatorics. Ann. of Math. (2) 164(1), 231–264 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bobenko, A.I., Suris, Y.: On discretization principles for differential geometry. The geometry of spheres. Russ. Math. Surv. 62(1), 1–43 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bobenko, A.I., Pottmann, H., Wallner, J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348(1), 1–24 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bohle, C., Peters, G.P.: Bryant surfaces with smooth ends. Commun. Anal. Geom. 17(4), 587–619 (2009). arXiv:math/0411480v2 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bohle, C.: Constant mean curvature tori as stationary solutions to the Davey–Stewartson equation. Math. Z. 217, 489–498 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bryant, R.L.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 23–53 (1984)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Bryant, R.L.: Surfaces of mean curvature one in hyperbolic space, Théorie des variétés minimales et applications (Palaiseau, 1983–1984). Astérisque 154–155, 321–347 (1987)Google Scholar
  15. 15.
    Calapso, P.: Sulle superficie a linee di curvatura isoterme. Rendiconti Circolo Matematico di Palermo 17, 275–286 (1903)Google Scholar
  16. 16.
    Calapso, P.: Sulle trasformazioni delle superficie isoterme. Ann. Mat. Pura Appl. 24, 11–48 (1915)zbMATHCrossRefGoogle Scholar
  17. 17.
    Carfì, D., Musso, E.: T-transformations of Willmore isothermic surfaces. Rend. Sem. Mat. Messina Ser. II, 69–86 (2000)Google Scholar
  18. 18.
    Cartan, E., Sur le problème général de la déformation, C. R. Congrés Strasbourg, 397–406; or Oeuvres Complètes, III 1, 539–548 (1920)Google Scholar
  19. 19.
    Cecil, T.E.: Lie sphere geometry: with applications to submanifolds. Springer, New York (1992)zbMATHCrossRefGoogle Scholar
  20. 20.
    Griffiths, P.A.: On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41, 775–814 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jensen, G.R.: Deformation of submanifolds of homogeneous spaces. J. Differ. Geom. 16, 213–246 (1981)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Jensen, G.R., Musso, E., Nicolodi, L.: Surfaces in classical geometries: a treatment by moving frames. Springer, New York (2016)Google Scholar
  23. 23.
    Hertrich-Jeromin, U., Musso, E., Nicolodi, L.: Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space. Ann. Glob. Anal. Geom. 19, 185–205 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hertrich-Jeromin, U.: Introduction to Möbius Differential Geometry, London Mathematical Society Lecture Note Series, 300. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  25. 25.
    Kobayashi, O.: Maximal surfaces in the 3-dimensional Minkowski space \(L^{3}\). Tokyo J. Math. 6(2), 297–309 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lawson, H.B.: Complete minimal surfaces in \(S^{3}\). Ann. Math. (2) 92, 335–374 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lee, S.: Spacelike surfaces of constant mean curvature \(\pm 1\) in de Sitter 3-space \({{\mathbb{S}}}^3_1(1)\). Ill. J. Math. 49(1), 63–98 (2005)zbMATHGoogle Scholar
  28. 28.
    Li, T., Wang, C.P.: Laguerre geometry of hypersurfaces in \(\mathbb{R}^{n}\). Manuscr. Math. 122, 73–95 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Musso, E., Nicolodi, L.: \(L\)-minimal canal surfaces. Rend. Mat. 15, 421–445 (1995)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Musso, E., Nicolodi, L.: A variational problem for surfaces in Laguerre geometry. Trans. Am. Math. Soc. 348, 4321–4337 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Musso, E., Nicolodi, L.: Isothermal surfaces in Laguerre geometry. Boll. Un. Mat. Ital. (7) II-B, Suppl. fasc. 2, 125–144 (1997)Google Scholar
  32. 32.
    Musso, E., Nicolodi, L.: On the equation defining isothermic surfaces in Laguerre geometry. In: Szenthe, J. (ed.) New Developments in Differential Geometry, Budapest, pp. 285–294. Kluwer Academic Publishers, Dordrecht (1996)Google Scholar
  33. 33.
    Musso, E., Nicolodi, L.: Laguerre geometry of surfaces with plane lines of curvature. Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Musso, E., Nicolodi, L.: The Bianchi–Darboux transform of \(L\)-isothermic surfaces. Int. J. Math. 11(7), 911–924 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Musso, E., Nicolodi, L.: Deformation and applicability of surfaces in Lie sphere geometry. Tohoku Math. J. (2) 58(2), 161–187 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Musso, E., Nicolodi, L.: Conformal deformation of spacelike surfaces in Minkowski space. Houst. J. Math. 35(4), 1029–1049 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Palmer, B.: Remarks on a variational problem in Laguerre geometry. Rend. Mat. Appl. (7) 19(2), 281–293 (1999)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Palmer, B.: Spacelike constant mean curvature surfaces in pseudo-Riemannian space forms. Ann. Glob. Anal. Geom. 8, 217–226 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Pottmann, H., Peternell, M.: Applications of Laguerre geometry in CAGD. Comput. Aided Geom. Des. 15(2), 165–186 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Pottmann, H., Grohs, P., Mitra, N.J.: Laguerre minimal surfaces, isotropic geometry and linear elasticity. Adv. Comput. Math. 31(4), 391–419 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Szereszewski, A.: \(L\)-isothermic and \(L\)-minimal surfaces. J. Phys. A 42(11), 115203–115217 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Skopenkov, M., Pottmann, H., Grohs, P.: Ruled Laguerre minimal surfaces. Math. Z. 272(1–2), 645–674 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Song, Y.-P.: Laguerre isothermic surfaces in \({{\mathbb{R}}}^3\) and their Darboux transformation. Sci. China Math. 56(1), 67–78 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Thomsen, G.: Über konforme Geometrie I: Grundlagen der konformen Flächentheorie. Hamb. Math. Abh. 3, 31–56 (1923)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Umehara, M., Yamada, K.: A parametrization of the Weierstrass formulae and perturbation of complete minimal surfaces in \({\mathbb{R}}^3\) into the hyperbolic 3-space. J. Reine Angew. Math. 432, 93–116 (1992)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Voss, K.: Verallgemeinerte Willmore-Flächen. Mathematisches Forschungsinstitut Oberwolfach, Workshop Report 42, 22–23 (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di ParmaParmaItaly

Personalised recommendations