Hyperbolic metric on the punctured disk ($ds=|dz|/(|z|\log (1/|z|))$)

The universal covering map $\tau \mapsto e^{i\tau }$ from the complex upper half-plane to the punctured unit disc pulls back the metric $|dz|/(|z|\log (1/|z|))$ to $|d\tau |/\mathrm{Im}\tau$. This is its complete curvature-minus-one hyperbolic metric, which is different from restricting the ordinary disk metric to the punctured disk.

// Destination: analysis.bigb

The curvature-minus-one hyperbolic metric on the punctured disk assigns distance $2\log n$ along the radial geodesic from $e^{-1}$ to $e^{-n^2}$. Its exponential distances therefore sum to a finite value; the triangle inequality gives the same conclusion for any fixed basepoint. Nevertheless no nonzero bounded holomorphic function has these zeros: the Riemann removable singularity theorem extends it at zero, and the identity theorem then makes it identically zero. Simple connectivity is necessary in the hyperbolic characterization of bounded analytic zero sets.