Results in Mathematics

, Volume 59, Issue 3–4, pp 545–562 | Cite as

Hyperbolic Relative Hyperspheres with Li-normalization

  • Min XiongEmail author
  • Baoying Yang


Consider a graph hypersurface
$$M=\{(x_1,\ldots,x_n,f(x_1,\ldots,x_n))\;\;|\;\; (x_1,\ldots,x_n)\in \Omega\}$$
where f is a strictly convex function defined on a convex domain Ω in real affine space A n . Assume that the hypersurface has a Li-normalization. We study hyperbolic affine hyperspheres with respect to this relative normalization and classify the subclass which is Euclidean complete.

Mathematics Subject Classification (2010)

53A15 35J60 35J65 53C42 58J60 


Euclidean completeness classification of hyperbolic Li hyperspheres Monge-Ampère equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cheng S.Y., Yau S.T.: Complete affine hypersurfaces, partI. The completeness of affine metrics. Commun. Pure and Appl. Math. 39, 839–866 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Gigena S.: On a conjecture of E. Calabi. Geom. Dedicata 11, 387–396 (1981)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Lazer A.C., McKenna P.J.: On singular boundary value problems for the Monge-Ampère operator. J. Math. Anal. Appl. 197, 341–362 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Li A.M.: Calabi conjecture on hyperbolic affine hyperspheres. Math. Z. 203, 483–491 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Li A.M.: Calabi conjecture on hyperbolic affine hyperspheres (2). Math. Annalen 293, 485–493 (1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    Li A.M., Simon U., Zhao G.S.: Global affine differential geometry of hypersurfaces. Walter de Gruyter, Berlin, New York (1993)CrossRefzbMATHGoogle Scholar
  7. 7.
    Li, A.M., Xu, R.W., Simon, U., Jia, F.: Affine Bernstein problems and Monge-Ampère equations. World Scientific (2010)Google Scholar
  8. 8.
    Sasaki T.: Hyperbolic affine hyperspheres. Nagoya Math. J. 77, 107–123 (1980)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Schirokow P.A., Schirokow A.P.: Affine differentialgeometrie. Teubner, Leipzig (1962)zbMATHGoogle Scholar
  10. 10.
    Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the affine differential geometry of hypersurfaces. Lecture notes, Science University Tokyo, (1991)Google Scholar
  11. 11.
    Wang B.F., Li A.M.: The Euclidean complete affine hypersurfaces with negative constant affine mean curvature. Results Math. 52, 383–398 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Wu, Y., Zhao, G.: Hypersurfaces with Li-normalization and prescibed Gauss-Kronecker curvature. Results Math. (this volume)Google Scholar
  13. 13.
    Xu R.W.: Bernstein properties for some relative parabolic affine hyperspheres. Results Math. 52, 409–422 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Jia F., Li A.M.: Interior estimates for solutions of a fourth order nonlinear partial differential equation. Diff. Geom. Appl. 25, 433–451 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Li A.M., Jia F.: A Bernstein properties of some fourth order partial differential equations. Results Math. 56, 109–139 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Li A.M., Xu R.: A cubic form differential inequality with applications to affine Kähler-Ricci flat manifolds. Results Math. 54, 329–340 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Li A.M., Xu R.: A rigidity theorem for affine Kähler-Ricci flat graph. Results Math. 56, 141–164 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Xu R., Li A.M., Li X.X.: Euclidean complete α relative extermal hypersurfaces. Sichuan Daxue Xuebao. 46, 1217–1223 (2009)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.College of MathematicsSichuan UniversityChengduPeople’s Republic of China
  2. 2.College of ScienceSouthwest University of Science and TechnologyMianyangPeople’s Republic of China
  3. 3.College of MathematicsSouthwest Jiaotong UniversityChengduPeople’s Republic of China

Personalised recommendations