Tracking a moving point in the plane
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- by Frederick P. Gardiner and Nikola Lakic PDF
- Trans. Amer. Math. Soc. 365 (2013), 1957-1975 Request permission
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 \[ \mathbb {K}= \Psi ^{-1}([identity])\] 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:
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$Teich_{\mathbb {K}}$ is the restriction of the Teichmüller infinitesimal metric on $Teich(R-p)$ to the submanifold $\mathbb {K}, \textrm {\ and \ }$
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$Kob_{\mathbb {K}}$ 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 \[ \lambda (p)=\inf \{||{\overline {\partial }}V||_{\infty }\},\] where the infimum is taken over all continuous functions $V$ for which $V(p)=1$ and $V$ vanishes on the boundary of $R,$ and \[ \rho (p)=\inf \{1/|f’(0)|\},\] where the infimum is taken over all holomorphic functions $f$ mapping the unit disc into $R$ with $f(0)=p.$ We also show
\[ (1/2) Kob_{\mathbb {K}} \leq Teich_{\mathbb {K}} \leq Kob_{\mathbb {K}} \textrm {\ \ and \ }\] \[ (1/2) \rho \leq \lambda \leq \rho ,\] 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
- MR Author ID: 198854
- Email: frederick.gardiner@gmail.com
- Nikola Lakic
- Affiliation: Department of Mathematics, Herbert H. Lehman College, Bronx, New York 10468
- Email: nlakic@lehman.cuny.edu
- 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
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 1957-1975
- MSC (2010): Primary 30F60; Secondary 32G15, 30C70, 30C75
- DOI: https://doi.org/10.1090/S0002-9947-2012-05757-6
- MathSciNet review: 3009650