Spherical conic metrics and realizability of branched covers

TL;DR

This paper links spherical conic metrics to the Hurwitz existence problem, providing a new method to identify unrealizable branched cover data and discovering infinite sets of such exceptional cases.

Contribution

It introduces a novel approach connecting spherical conic metrics to branched cover realizability, enabling the identification of new exceptional data sets.

Findings

New infinite sets of exceptional branched cover data found

A method to determine unrealizability of certain branched covers

Enhanced understanding of the connection between metrics and combinatorial data

Abstract

Branched covers between Riemann surfaces are associated with certain combinatorial data, and Hurwitz existence problem asks whether given data satisfying those combinatorial constraints can be realized by some branched cover. We connect recent development in spherical conic metrics to this old problem, and give a new method of finding exceptional (unrealizable) branching data. As an application, we find new infinite sets of exceptional branched cover data on the Riemann sphere.