On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

Yan Wu,Zunwei Fu,Yi Qi

doi:10.1155/2015/276719

Yan Wu, Zunwei Fu + Show 1 more

Open Access

https://doi.org/10.1155/2015/276719

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Abstract

LetAZ(R)be the infinitesimal asymptotic Teichmüller space of a Riemann surfaceRof infinite type. It is known thatAZ(R)is the quotient Banach space of the infinitesimal Teichmüller spaceZ(R), whereZ(R)is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points inAZ(R). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller spaceAZ(D)by constructing a special degenerating sequence.

Highlights

Let R be a hyperbolic Riemann surface, that is, a Riemann surface with universal covering surface which is conformally equivalent to the open unit disk D
The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points in AZ(R)
We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmuller space AZ(D) by constructing a special degenerating sequence

Summary

Introduction

Let R be a hyperbolic Riemann surface, that is, a Riemann surface with universal covering surface which is conformally equivalent to the open unit disk D. Is the set of all infinitesimal asymptotic equivalence classes of μ’s in L∞(R). Given two points in infinitesimal Teichmuller space Z(R), it is shown in [6] that there exists at least one geodesic joining them. It is proved in [14] that there exists precisely one geodesic segment joining the basepoint [0]z and [μ]z in Z(R) if and only if [μ]z contains a uniquely extremal Beltrami coefficient with constant modulus. We will prove the nonuniqueness of geodesics in the universal infinitesimal asymptotic Teichmuller space AZ(D) by constructing a special degenerating sequence {φn} in Q(D).

Preliminaries

Constructing a Special Sequence Degenerating towards a Boundary Point

Nonuniqueness of Geodesics Joining Every