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On convergence to a football

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Abstract

We show that spheres of positive constant curvature with n (\(n\ge 3\)) 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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Acknowledgments

Both authors would like to thank Jian Song and Lihe Wang for discussion. Both authors thank the referee for useful comments. Part of the work was done when both authors were visiting Beijing International Center for Mathematical Research. We are thankful for its hospitality.

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Authors and Affiliations

  1. University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242, USA

    Hao Fang

  2. Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240, China

    Mijia Lai

Authors
  1. Hao Fang
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  2. Mijia Lai
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Correspondence to Hao Fang.

Additional information

H.F.’s work is partially supported by Simons Foundation and NSF DMS-100829. M.L.’s work is partially supported by Shanghai Yangfan 15YF1406200 and NSFC No. 11501360.

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Fang, H., Lai, M. On convergence to a football. Math. Ann. 366, 83–100 (2016). https://doi.org/10.1007/s00208-015-1303-z

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  • DOI: https://doi.org/10.1007/s00208-015-1303-z

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