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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)



Tracking a moving point in the plane

Authors: Frederick P. Gardiner and Nikola Lakic
Journal: Trans. Amer. Math. Soc. 365 (2013), 1957-1975
MSC (2010): Primary 30F60; Secondary 32G15, 30C70, 30C75
Published electronically: September 18, 2012
MathSciNet review: 3009650
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Abstract: The Teichmüller theory of any hyperbolic Riemann surface $ R$ induces two closely related metrics on $ R$ in the following way. From a theorem of Bers, the fiber

$\displaystyle \mathbb{K}= \Psi ^{-1}([identity])$

of the forgetful map $ \Psi $ from the Teichmüller space $ Teich(R-p)$ onto the Teichmüller space $ Teich(R)$ is conformal to a disc and the evaluation map $ \mathbb{K} \ni [f] \mapsto f(p) \in R$ is a universal covering of $ R.$ There are two infinitesimal metrics on $ \mathbb{K}$ coming from Kobayashi's construction:
  1. $ Teich_{\mathbb{K}}$ is the restriction of the Teichmüller infinitesimal metric on
    $ Teich(R-p)$ to the submanifold $ \mathbb{K}, {\rm\ and \ }$
  2. $ Kob_{\mathbb{K}}$ is the Kobayashi metric on $ \mathbb{K}.$

We show these metrics, respectively, are the lifts via the evaluation map of infinitesimal forms $ \lambda $ and $ \rho $ on $ R,$ where $ \lambda $ and $ \rho $ are the Teichmüller and Poincaré densities. $ \lambda $ and $ \rho $ have very different descriptions. For plane domains

$\displaystyle \lambda (p)=\inf \{\vert\vert{\overline {\partial }}V\vert\vert _{\infty }\},$

where the infimum is taken over all continuous functions $ V$ for which $ V(p)=1$ and $ V$ vanishes on the boundary of $ R,$ and

$\displaystyle \rho (p)=\inf \{1/\vert f'(0)\vert\},$

where the infimum is taken over all holomorphic functions $ f$ mapping the unit disc into $ R$ with $ f(0)=p.$ We also show

$\displaystyle (1/2) Kob_{\mathbb{K}} \leq Teich_{\mathbb{K}} \leq Kob_{\mathbb{K}} {\rm\ \ and \ }$

$\displaystyle (1/2) \rho \leq \lambda \leq \rho ,$

and $ \lambda /\rho =1/2$ when $ R$ is simply connected, $ \lambda /\rho =1$ when $ R$ is a thrice punctured sphere, and in all other cases these inequalities are strict.

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Additional Information

Frederick P. Gardiner
Affiliation: Department of Mathematics, Graduate School and University Center of CUNY, New York, New York 10016 – and – Department of Mathematics, Brooklyn College, CUNY, Brooklyn, New York 11210

Nikola Lakic
Affiliation: Department of Mathematics, Herbert H. Lehman College, Bronx, New York 10468

Received by editor(s): November 10, 2009
Received by editor(s) in revised form: July 20, 2011
Published electronically: September 18, 2012
Additional Notes: The second author was partially supported by NSF grant 0700052
Article copyright: © Copyright 2012 American Mathematical Society