Complex rigidity of Teichm\"uller spaces
Samuel L. Krushkal
TL;DR
This paper reviews classical and recent results on the geometric properties of Teichm"uller spaces, focusing on their shape and biholomorphic equivalences in complex analysis.
Contribution
It summarizes known and new findings regarding the starlikeness and convexity properties of Teichm"uller spaces in complex Banach spaces.
Findings
Teichm"uller spaces are not starlike in the Bers embedding.
Higher-dimensional Teichm"uller spaces are not biholomorphically equivalent to bounded convex domains.
The paper discusses unresolved problems and recent advances in the field.
Abstract
We outline old and new results concerning the well-known problems in the Teichm\"uller space theory, i.e., whether these spaces are starlike in the Bers holomorphic embedding and whether any Teichm\"uller space of dimension greater than 1 is biholomorhically equivalent to bounded convex domain in a complex Banach space.
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Taxonomy
TopicsAnalytic and geometric function theory · Holomorphic and Operator Theory