The complex structure of the Teichmüller space of circle diffeomorphisms in the Zygmund smooth class II

IF 1.2 3区 数学 Q1 MATHEMATICS
Katsuhiko Matsuzaki
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引用次数: 0

Abstract

In our previous paper with the same title, we established the complex Banach manifold structure for the Teichmüller space of circle diffeomorphisms whose derivatives belong to the Zygmund class. This was achieved by demonstrating that the Schwarzian derivative map is a holomorphic split submersion. We also obtained analogous results for the pre-Schwarzian derivative map. In this second part of the study, we investigate the structure of the image of the pre-Schwarzian derivative map, viewing it as a fiber space over the Bers embedding of the Teichmüller space, and prove that it forms a real-analytic disk-bundle. Furthermore, we consider the little Zygmund class and establish corresponding results for the closed Teichmüller subspace consisting of mappings in this class. Finally, we construct the quotient space of this subspace in analogy with the asymptotic Teichmüller space and prove that the quotient Bers embedding and pre-Bers embedding are well-defined and injective, thereby endowing it with a complex structure modeled on a quotient Banach space.
Zygmund光滑类中圆微分同态的teichm ller空间的复结构
在上一篇同名论文中,我们建立了导数属于Zygmund类的圆微分同胚的teichm空间的复Banach流形结构。这是通过证明Schwarzian导数映射是一个全纯分裂淹没来实现的。对于pre-Schwarzian导数图,我们也得到了类似的结果。在本研究的第二部分中,我们研究了pre-Schwarzian导数映射的图像结构,将其看作是在teichm ller空间的Bers嵌入上的纤维空间,并证明了它形成了一个实解析盘束。进一步,我们考虑了小Zygmund类,并建立了该类中由映射组成的闭teichm ller子空间的相应结果。最后,我们类比渐近的teichm ller空间构造了该子空间的商空间,并证明了商Bers的嵌入和pre-Bers的嵌入是定义良好的内射,从而赋予其以商Banach空间为模型的复结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.50
自引率
7.70%
发文量
790
审稿时长
6 months
期刊介绍: The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.
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