On convergence to a football

TL;DR

This paper demonstrates that conic spheres with positive constant curvature and multiple conic points converge to a football-shaped sphere with two conic points in the Gromov-Hausdorff topology, using geometric and analytical methods.

Contribution

It provides a rigorous proof of convergence of conic spheres to a football shape, employing both geometric descriptions and PDE analysis.

Findings

Convergence of conic spheres to a football shape in Gromov-Hausdorff topology.

Two different proofs: geometric via convex polytopes, analytical via PDE estimates.

Establishment of a priori estimates for the conformal factors.

Abstract

We show that spheres of positive constant curvature with $n$ ($n\geq3$) conic points converge to a sphere of positive constant curvature with two conic points (or called an (American) football) in Gromov-Hausdorff topology when the corresponding singular divisors converge to a critical divisor in the sense of Troyanov. We prove this convergence in two different ways. Geometrically, the convergence follows from Luo-Tian's explicit description of conic spheres as boundaries of convex polytopes in $S^{3}$. Analytically, regarding the conformal factors as the singular solutions to the corresponding PDE, we derive the required a priori estimates and convergence result after proper reparametrization.