On convergence to a football

Hao Fang, Mijia Lai

Published 2015-01-27Version 1

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 an (American) football in Gromov-Hausdorff topology when the conic angles of the sequence pass from the subcritical case in the sense of Troyanov to the critical case in the limit. 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, considering the conformal factors as the singular solutions to the corresponding PDE, we derive the required a priori estimates and convergence result after proper reparametrization.