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Some Geometric Aspects of the Hessian One Equation

  • Antonio Martínez
  • Francisco Milán
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

Abstract

The Hessian one equation and its complex resolution provides an important tool in the study of improper affine spheres in \(\mathbb{R}^{3}\) with some kind of singularities. The singular set can be characterized and, in most of the cases, it determines the surface. Here, we show how to obtain improper affine spheres with a prescribed singular set and construct some global examples with the desired singularities. We also classify improper affine spheres admitting a planar singular set.

Keywords

Conformal Structure Entire Solution Complex Resolution Singular Curve Isolate Singularity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Aledo, J.A., Chaves R.M.B., Gálvez,J.A.: The cauchy problem for improper affine spheres and the Hessian one equation. Trans. Am. Math. Soc. 359, 4183–4208 (2007)CrossRefMATHGoogle Scholar
  2. 2.
    Aledo, J.A., Espinar J.M., Gálvez, J.A.: The Codazzi equation for surfaces. Adv. Math. 224, 2511–2530 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Calabi, E.: Affine differential geometry and holomorphic curves. Lect. Notes Math. 1422, 15–21 (1990)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Chaves, R.M.B., Dussan, M.P., Magid, M.: Björling problem for timelike surfaces in the Lorentz-Minkowski space. J. Math. Anal. Appl. 377(1), 481–494 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cortés, V., Lawn, M.A., Schäfer, L.: Affine hyperspheres associated to special para-Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 3(5–6), 995–1009 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Craizer, M.: Singularities of convex improper affine maps. J. Geom. 103(2), 207–217 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Ferrer, L., Martínez, A., Milán, F.: Symmetry and uniqueness of parabolic affine spheres. Math. Ann. 305, 311–327 (1996)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Ferrer, L., Martínez, A., Milán, F.: An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres. Math. Z. 230, 471–486 (1999)MATHGoogle Scholar
  9. 9.
    Gálvez, J.A., Martínez, A., Mira, P.: The space of solutions to the Hessian one equation in the finitely punctured plane. J. Math. Pures Appl. 84, 1744–1757 (2005)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Huber, A.: On subharmonics functions and differential geometry in the large. Comment. Math. Helv. 32, 13–72 (1957)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Inoguchi, J., Toda, M.: Timelike minimal surfaces via loop groups. Acta Appl. Math. 83(3), 313–355 (2004)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Ishikawa, G., Machida, Y.: Singularities of improper affine spheres and surfaces of constant Gaussian curvature. Int. J. Math. 17(39), 269–293 (2006)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Jörgens, K.: Über die Lösungen der differentialgleichung \(rt - s^{2} = 1\). Math. Ann. 127, 130–134 (1954)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Jörgens, K.: Harmonische Abbildungen und die Differentialgleichung \(rt - s^{2} = 1\). Math. Ann. 129, 330–344 (1955)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kawakami, Y., Nakajo, D.: Value distribution of the Gauss map of improper affine spheres. J. Math. Soc. Jpn. 64 no. 3, 799–821 (2012)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Klotz Milnor, T.: Codazzi pairs on surfaces, Global differential geometry and global analysis. Proc. Colloq. Berlin 1979. Lect. Notes Math. 838, 263–274 (1981)Google Scholar
  17. 17.
    Kokubu, M., Rossman, W., Saji, K., Umehara, M., Yamada, K.: Singularities of flat fronts in hyperbolic space. Pac. J. Math. 221, 303–351 (2005)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Li, A.M., Jia, F., Simon U., Xu, R.: Affine Bernstein Problems and Monge-Ampère Equations. World Scientific, Singapore (2010)CrossRefMATHGoogle Scholar
  19. 19.
    Li, A.M., Simon, U., Zhao, G.: Global Affine Differential Geometry of Hypersurfaces. Walter de Gruyter, Berlin (1993)CrossRefMATHGoogle Scholar
  20. 20.
    Loftin, J.C.: Survey on affine spheres. Handbook of Geometric Analysis. Adv. Lect. Math., 13(2), 161–192; International Press, Somerville (2010)Google Scholar
  21. 21.
    Martínez, A.: Improper affine maps. Math. Z. 249, 755–766 (2005)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Martínez, A., Milán, F.: Flat fronts in the hyperbolic 3-space with prescribed singularities. Ann. Glob. Anal. Geom. 46(3), 227–239 (2014)CrossRefMATHGoogle Scholar
  23. 23.
    Milán, F.: Singularities of improper affine maps and their Hessian equation. Adv. Math. 251, 22–34 (2014)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Milán, F.: The Cauchy problem for indefinite improper affine spheres and their Hessian equation. J. Math. Anal. Appl. 405, 183–190 (2013)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Nakajo, D.: A representation formula for indefinite improper affine spheres. Results Math. 55, 139–159 (2009)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  27. 27.
    Trudinger, N.S., Wang, X.J.: The Monge-Ampère equation and its geometric applications. Handbook of Geometric Analysis. Adv. Lect. Math. 7(1), 467–524; International Press, Somerville (2008)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.University of GranadaGranadaSpain

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