数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (5): 1173-1181.doi: 10.1016/S0252-9602(09)60095-3

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ON THE CONVERGENCE OF CIRCLE PACKINGS TO THE QUASICONFORMAL MAP

 黄小军, 沈良   

  1. College of Mathematics and Physics, Chongqing University, Chongqing 400044, China;Institute of Mathematics, Academiy of Mathematics &|System Sciences, Chinese Academiy of Sciences, Beijing 100190, China
  • 收稿日期:2008-06-24 出版日期:2009-09-20 发布日期:2009-09-20
  • 基金资助:

    This work was supported by the  National Natural Science Foundation of China (10701084) and  Chongqing Natural Science Foundation (2008BB0151)

ON THE CONVERGENCE OF CIRCLE PACKINGS TO THE QUASICONFORMAL MAP

 HUANG Xiao-Jun, CHEN Liang   

  1. College of Mathematics and Physics, Chongqing University, Chongqing 400044, China;Institute of Mathematics, Academiy of Mathematics &|System Sciences, Chinese Academiy of Sciences, Beijing 100190, China
  • Received:2008-06-24 Online:2009-09-20 Published:2009-09-20
  • Supported by:

    This work was supported by the  National Natural Science Foundation of China (10701084) and  Chongqing Natural Science Foundation (2008BB0151)

摘要:

Rodin and Sullivan (1987) proved Thurston's conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded
simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to ∞.

关键词: circle packing, quasiconformal map, complex dilation

Abstract:

Rodin and Sullivan (1987) proved Thurston's conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded
simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to ∞.

Key words: circle packing, quasiconformal map, complex dilation

中图分类号: 

  • 30C85