Chinese Quarterly Journal of Mathematics ›› 2024, Vol. 39 ›› Issue (2): 180-184.doi: 10.13371/j.cnki.chin.q.j.m.2024.02.006

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The Gauss-Bonnet Formula of a Conical Metric on a Compact Riemann Surface

FANG Han-bing1, XU Bin2, YANG Bai-rui3   

  1. 1. Mathematics Department, Stony Brook University, NY 11794, United States; 2. CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China; 3. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
  • Received:2023-02-09 Online:2024-06-30 Published:2024-06-30
  • Contact: YANG Bai-rui (1998-), native of Nanjing, Jiangsu, master student of University of Science and Technology of China, engages in geometric analysis. E-mail: bry@mail.ustc.edu.cn
  • About author:FANG Han-bing (1998-), male, native of Chizhou, Anhui, PhD student of Stony Brook University, engages in differential geometry; XU Bin (1976-), male, native of Yiyang, Hunan, associate professor of University of Science and Technology of China, engages in differential geometry; YANG Bai-rui (1998-), native of Nanjing, Jiangsu, master student of University of Science and Technology of China, engages in geometric analysis.
  • Supported by:
    Support by the Project of Stable Support for Youth Team in Basic Research Field, CAS (Grant No. YSBR-001) and NSFC (Grant Nos. 12271495, 11971450 and 12071449).

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Abstract:  We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric. We also construct explicitly some conical metrics whose curvature is not integrable.

Key words:  Gauss-Bonnet formula, Conical metric, Riemann surface, Gaussian curvature; Lebesgue integrable

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