数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (3): 765-781.doi: 10.1016/S0252-9602(16)30038-8

• 论文 • 上一篇    下一篇

THE ASSOCIATED FAMILIES OF SEMI-HOMOGENEOUS COMPLETE HYPERBOLIC AFFINE SPHERES

林至诚1, 王二小2   

  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
  • 收稿日期:2015-03-31 修回日期:2015-05-12 出版日期:2016-06-25 发布日期:2016-06-25
  • 通讯作者: Erxiao WANG,E-mail:maexwang@ust.hk E-mail:maexwang@ust.hk
  • 作者简介:Zhicheng LIN,E-mail:flyriverms@qq.com
  • 基金资助:

    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).

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).

摘要:

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.

关键词: hyperbolic affine spheres, isothermal coordinates, Weierstrass elliptic functions, Monge-Ampère equation, Tzitzéica equation

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

中图分类号: 

  • 37K25