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Schwarz reflection geometry. II: Local and global behavior of the exponential map. (English) Zbl 1132.53029

Summary: A local normal form is obtained for geodesics in the space \(\Lambda=\{\Gamma\}\) of analytic Jordan curves in the extended complex plane with symmetric space multiplication \(\Gamma_1\cdot\Gamma_2\) defined by Schwarzian reflection of \(\Gamma_2\) in \(\Gamma_1\). Local geometric features of \((\Lambda, \cdot)\) will be seen to reflect primarily the structure of the Witt algebra, while issues of global behavior of the exponential map will be viewed in the context of conformal mapping theory.
For Part I see the authors, Math. Z. 244, No. 4, 775–804 (2003; Zbl 1078.53044).

MSC:

53C35 Differential geometry of symmetric spaces
53A30 Conformal differential geometry (MSC2010)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Citations:

Zbl 1078.53044
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Full Text: DOI Euclid