Acta mathematica scientia,Series B ›› 2016, Vol. 36 ›› Issue (3): 765-781.doi: 10.1016/S0252-9602(16)30038-8

• Articles • Previous Articles     Next Articles

THE ASSOCIATED FAMILIES OF SEMI-HOMOGENEOUS COMPLETE HYPERBOLIC AFFINE SPHERES

Zhicheng LIN1, Erxiao WANG2   

  1. 1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China;
    2. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
  • Received:2015-03-31 Revised:2015-05-12 Online:2016-06-25 Published:2016-06-25
  • Contact: Erxiao WANG,E-mail:maexwang@ust.hk E-mail:maexwang@ust.hk
  • Supported by:

    The authors were supported by the NSF of China (10941002, 11001262), and the Starting Fund for Distinguished Young Scholars of Wuhan Institute of Physics and Mathematics (O9S6031001).

Abstract:

Hildebrand classified all semi-homogeneous cones in R3 and computed their cor-responding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we con-struct the whole associated family for each of Hildebrand's examples. The generic member of these affine spheres is given by Weierstrass P, ζ and σ functions. In general any regular convex cone in R3 has a natural associated S1-family of such cones, which deserves further studies.

Key words: hyperbolic affine spheres, isothermal coordinates, Weierstrass elliptic functions, Monge-Ampère equation, Tzitzéica equation

CLC Number: 

  • 37K25
Trendmd