# Construction of nearly hyperbolic distance on punctured spheres

## TL;DR

This paper introduces a new distance function on punctured spheres that approximates hyperbolic distance, providing a computationally simpler alternative with potential applications in complex analysis.

## Contribution

It constructs a nearly hyperbolic distance on punctured spheres and proposes an easier-to-compute comparable quantity, expanding tools for geometric analysis.

## Key findings

- The new distance is comparable with hyperbolic distance.
- The approach extends from punctured disks to n-punctured spheres.
- A simpler, comparable quantity is proposed for easier computation.

## Abstract

We define a distance function on the bordered punctured disk $0<|z|\le 1/e$ in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk $0<|z|<1.$ As an application, we will construct a distance function on an $n$-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute.