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1991•Complex Variables Theory and Application An International JournalRequires access

Nonuniqueness of geodesics in infinite dimensional teichmuller spaces

Zhong Li

Open publisher page 40 citations

Abstract

Two extremal Beltrami differentials μ0 and μ1 with the following properties are constructed: μ0 and μ1 represent a same point P in the universal Teichmüller space, while the paths tμ0 and tμ1. 0 ≤ t ≤ 1, represent different geodesics joining the original point to the point P. This answers an open problem on the uniqueness of geodesics in infinite dimensional Teichmiiller spaces proposed by E P. Gardiner. Moreover, it is shown that one can get infinitely many geodesis joining two points in such a way.

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What this paper is about

Two extremal Beltrami differentials μ0 and μ1 with the following properties are constructed: μ0 and μ1 represent a same point P in the universal Teichmüller space, while the paths tμ0 and tμ1. 0 ≤ t ≤ 1, represent different geodesics joining the original point to the point P. This answers an open problem on the uniqueness of geodesics in infinite dimensional Teichmiiller spaces proposed by E P. Gardiner. Moreover, it is shown that one can get infinitely many geodesis joining two points in such a way.

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Available abstract

Two extremal Beltrami differentials μ0 and μ1 with the following properties are constructed: μ0 and μ1 represent a same point P in the universal Teichmüller space, while the paths tμ0 and tμ1. 0 ≤ t ≤ 1, represent different geodesics joining the original point to the point P. This answers an open problem on the uniqueness of geodesics in infinite dimensional Teichmiiller spaces proposed by E P. Gardiner. Moreover, it is shown that one can get infinitely many geodesis joining two points in such a way.

Key concepts: Geodesic, Mathematics, Uniqueness, Point (geometry), Pure mathematics, Teichmüller space, Space (punctuation), Mathematical analysis

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