Extremal plane quasiconformal mappings with given boundary values
Extremal plane quasiconformal mappings with given boundary values
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DOI：
10.1090/s0002-9904-1973-13232-x
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发表时间：
1973-03
影响因子：
1.3
通讯作者：
E. Reich;K. Strebel
中科院分区：
文献类型：
--
作者：
E. Reich;K. Strebel
出版商摘要 本章讨论单位圆盘 E: |z| 的拟共形自映射 f = f x x 表示映射 f 的复膨胀。通过延拓，f 导致边界 ∂E 与其自身同胚。本章基于长度面积法证明了单位圆盘的汉密尔顿定理。由具有 n 个不同边界点的单位圆盘的极值拟共形映射确定的二次微分 φ n 形成该积分的最大化序列。本章讨论了一个悬而未决的问题，即条件（*）是否充分，换句话说，汉密尔顿定理的逆命题是否成立。本章中提出的证明仅使用了长度面积方法和有限范数二次微分的众所周知的性质。
Publisher Summary This chapter discusses the quasiconformal self mappings f = f x of the unit disc E: |z| x denotes the complex dilatation of the mapping f . By continuation, f induces a homeomorphism of the boundary ∂E onto itself. The chapter presents a proof of Hamilton's theorem for the unit disc, based on the length-area method. The quadratic differentials φ n determined by the extremal quasiconformal mappings of the unit disc with n distinguished boundary points form a maximizing sequence for this integral. The chapter discusses an open problem regarding whether the condition (*) is also sufficient, in other words, whether the converse of Hamilton's theorem is true. The proof presented in the chapter makes use only of the length-area method and well-known properties of quadratic differentials with finite norm.