Tracking a moving point in the plane

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:

1. $Teich_{\mathbb {K}}$ is the restriction of the Teichmüller infinitesimal metric on $Teich(R-p)$ to the submanifold $\mathbb {K}, \textrm {\ and \ }$

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 \[ \lambda (p)=\inf \{||{\overline {\partial }}V||_{\infty }\},\] 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.