Spherical surfaces with conical points: systole inequality and moduli spaces with many connected components

TL;DR

This paper investigates the structure of the moduli space of spherical surfaces with conical points, establishing conditions for non-emptiness, connectedness, and a systole inequality linking geometric and conformal invariants.

Contribution

It provides new insights into the topology and geometry of moduli spaces of spherical metrics with conical singularities, including an explicit systole inequality and analysis of the forgetful map.

Findings

Conditions for non-emptiness and connectedness of the moduli space

An explicit systole inequality relating metric and conformal invariants

Properness of the forgetful map from spherical to Riemann surface moduli

Abstract

In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical surfaces with conical points to the associated moduli space of pointed Riemann surfaces, such as its properness, which follows from an explicit systole inequality that relates metric invariants (spherical systole) and conformal invariant (extremal systole).