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arXiv · 1501.06881

On convergence to a football

Hao Fang·Mijia Lai
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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.

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Hao Fang, Mijia Lai. 2015-10-06. On convergence to a football. https://arxiv.org/abs/1501.06881

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