On curvature pinching of conic 2-spheres

TL;DR

This paper investigates the curvature pinching of conic 2-spheres without Einstein metrics, establishing optimal constants and describing their Gromov-Hausdorff limits, extending previous results to more general conic configurations.

Contribution

It determines the best curvature pinching constant for positive curvature conic 2-spheres and characterizes their limits, generalizing prior work to multiple conic points.

Findings

Established the optimal curvature pinching constant for positive conic 2-spheres.

Identified the Gromov-Hausdorff limits when the pinching constant is approached.

Extended previous results to conic 2-spheres with multiple conic points.

Abstract

We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a generalization of the previous results of Chen-Lin and Bartolucci for 2-spheres with one or two conic points.