[论文解读] Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller

Athanase Papadopoulos|arXiv (Cornell University)|Feb 13, 2017

History and Theory of Mathematics参考文献 57被引用 6

一句话总结

本文追溯了拟共形映射从古代地图学（始于托勒密的地理投影）到20世纪初蒂希穆勒奠基性工作的历史发展。文章强调了蒂索特引入的变形椭圆（indicatrix of distortion）概念，格罗茨施、拉夫连蒂耶夫、阿赫夫尔斯与蒂希穆勒在极值拟共形映射方面的发展，并展示了蒂希穆勒关于极值拟共形映射的存在性与唯一性定理如何成为蒂希穆勒理论与复分析的核心，特别是在通过二次微分解决比伯巴赫猜想方面。

ABSTRACT

The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasiconformality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. We then survey early quasiconformal theory in the works of Grötzsch, Lavrentieff, Ahlfors and Teichmüller, which are the 20th-century founders of the theory. The period we consider starts with Claudius Ptolemy (c. 100--170 A.D.) and ends with Oswald Teichmüller (1913--1943). We mention the works of several mathematicians-geographers done in this period, including Euler, Lagrange, Lambert, Gauss, Chebyshev, Darboux and others. We highlight in particular the work of Nicolas-Auguste Tissot (1824--1897), a French mathematician and geographer who (according to our knowledge) was the first to introduce the notion of a mapping which transforms infinitesimal circles into infinitesimal ellipses, studying parameters such as the ratio of the major to the minor axes of such an infinitesimal ellipse, its area divided by the area of the infinitesimal circle of which it is the image, and the inclination of its axis with respect to a fixed axis in the plane. We also give some information about the lives and works of Grötzsch, Lavrentieff, Ahlfors and Teichmüller. The latter brought the theory of quasiconformal mappings to a high level of development. He used it in an essential way in his investigations of Riemann surfaces and their moduli and in function theory (in particular, in his work on the Bieberbach conjecture and the type problem). We survey in detail several of his results. We also discuss some aspects of his life and writings, explaining why his papers were not read and why some of his ideas are still unknown even to Teichmüller theorists. The final version of this paper will appear in the book "Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds, and Picard-Fuchs Equations" (ed. L. Ji and S.-T. Yau), International Press and Higher Education Press (2017).

研究动机与目标

• 追溯拟共形映射在古代与早期现代地图学中的历史根源，特别是试图最小化球面投影中畸变的尝试。
• 突出此前被忽视的贡献，尤其是蒂索特对无穷小椭圆及其几何参数的系统研究，其时间早于正式的拟共形理论。
• 综述20世纪初格罗茨施、拉夫连蒂耶夫、阿赫夫尔斯与蒂希穆勒的奠基性工作，强调极值拟共形映射的出现。
• 解释蒂希穆勒关于极值映射与二次微分的工作如何为解决复分析中的重大问题（包括比伯巴赫猜想）提供了深刻框架。
• 主张蒂希穆勒未发表的洞见至今仍被低估，亟需学术界重新关注。

提出的方法

• 对从托勒密（约公元100–170年）到蒂索特（1824–1897年）的地理制图历史进行分析，聚焦于投影中畸变的数学处理。
• 阐述早期拟共形理论，包括格罗茨施在矩形之间构造极值映射的工作，以及拉夫连蒂耶夫在畸变最小化方面的贡献。
• 深入考察阿赫夫尔斯在推进理论方面的作用，以及他对蒂希穆勒重要性的认识。
• 对蒂希穆勒未发表论文进行详细分析，尤其是其1943年关于极值问题及在单芽函数论中使用二次微分的工作。
• 应用蒂希穆勒原理：极值映射由具有指定极点的二次微分决定，其极点的阶数对应于函数系数的归一化条件。
• 利用琴本1962年国际数学家大会演讲，重构并形式化蒂希穆勒的直觉性原理，转化为涉及二次微分与映射系数的精确不等式。

实验结果

研究问题

• RQ1 古代与早期现代地图制图实践如何为拟共形映射奠定概念基础？
• RQ2 蒂索特对无穷小椭圆（蒂索特椭圆）的研究在多大程度上预见了现代拟共形畸变理论？
• RQ3 格罗茨施在矩形之间构造极值映射的结果，如何推广为蒂希穆勒在黎曼曲面上拟共形映射的存在性与唯一性定理？
• RQ4 二次微分在蒂希穆勒解决比伯巴赫猜想的过程中发挥了何种作用？琴本如何在后续工作中形式化这一思想？
• RQ5 为何蒂希穆勒的思想（尤其是其未发表论文中的思想）在数学界长期未被广泛认知或充分重视？

主要发现

• 蒂索特是首位正式定义并研究投影下无穷小圆的畸变的人，引入了诸如像椭圆长轴与短轴之比、其面积相对于原圆的大小，以及椭圆主轴方向等参数。
• 蒂希穆勒1943年的论文提出了一个普遍原理：单芽函数论中极值问题的解由具有极点的二次微分决定，其极点的阶数对应于函数系数的归一化条件。
• 蒂希穆勒存在性与唯一性定理将格罗茨施关于矩形映射的结果推广至任意黎曼曲面，建立了拟共形映射与蒂希穆勒空间之间深刻的联系。
• 琴本1962年国际数学家大会演讲将蒂希穆勒原理形式化为一个精确不等式：当且仅当映射是二次微分所诱导度量下的等距映射时，等号成立。
• 尽管蒂希穆勒关于比伯巴赫猜想的工作未完全发表，但其已提供使用极值拟共形映射与二次微分解决该问题的基础技术，后由琴本及其他学者进一步发展。
• 尽管蒂希穆勒具有深刻洞见，但其论文因内容艰涩及所处时代的政治与历史背景，长期未被广泛阅读，导致其贡献被延迟认可。