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On the explicit expression of a conformal metric of constant curvature one near a conical singularity

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.06.001
  • Received Date: 26 November 2016
  • Rev Recd Date: 20 February 2017
  • Publish Date: 30 June 2017
    Abstract

  • Abstract

    Near a conical singularity with angle 2πα>0 of a conformal metric of constant curvature one, it was proved by using the developing map that there exists a suitable complex coordinate z under which the metric has the expression of 

    Abstract

    Near a conical singularity with angle 2πα>0 of a conformal metric of constant curvature one, it was proved by using the developing map that there exists a suitable complex coordinate z under which the metric has the expression of 
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    • References(10)
  • [1]
    MCOWEN R C. Point singularities and conformal metrics on Riemann surfaces[J]. Proc Amer Math Soc, 1988, 103(1): 222-224.
    [2]
    TROYANOV M. Prescribing curvature on compact surfaces with conical singularities[J]. Trans Amer Math Soc, 1991, 324(2): 793-821.
    [3]
    TROYANOV M. Metrics of constant curvature on a sphere with two conical singularities[C]// Differential Geometry. Berlin: Springer-Verlag, 1989: 296-306.
    [4]
    LUO F, TIAN G. Liouville equation and spherical convex polytopes[J]. Proc Amer Math Soc, 1992, 116(4): 1119-1129.
    [5]
    UMEHARA M, YAMADA K. Metrics of constant curvature 1 with three conical singularities on the 2-sphere[J]. Illinois J Math, 2000, 44(1): 72-94.
    [6]
    EREMENKO A. Metrics of positive curvature with conical singularities on the sphere[J]. Proc Amer Math Soc, 2004, 132(11): 3349-3355.
    [7]
    FURUTA M, HATTORI Y. 2-dimensional singular spherical space forms[Z]. manuscript, 1998.
    [8]
    FUJIMORI S, KAWAKAMI Y, KOKUBU M, et al. CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere[J]. Proc Japan Acad Ser A Math Sci, 2011, 87(8): 144-149.
    [9]
    CHEN Q, WANG W, WU Y, et al. Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surface[J]. Pacific Journal of Mathematics, 2015, 273: 75-100.
    [10]
    BRYANT R L. Surfaces of mean curvature one in hyperbolic space[J]. Theorie des Varietes Minimales et Applications, 1988, 154-155: 321-347.
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    [1]
    MCOWEN R C. Point singularities and conformal metrics on Riemann surfaces[J]. Proc Amer Math Soc, 1988, 103(1): 222-224.
    [2]
    TROYANOV M. Prescribing curvature on compact surfaces with conical singularities[J]. Trans Amer Math Soc, 1991, 324(2): 793-821.
    [3]
    TROYANOV M. Metrics of constant curvature on a sphere with two conical singularities[C]// Differential Geometry. Berlin: Springer-Verlag, 1989: 296-306.
    [4]
    LUO F, TIAN G. Liouville equation and spherical convex polytopes[J]. Proc Amer Math Soc, 1992, 116(4): 1119-1129.
    [5]
    UMEHARA M, YAMADA K. Metrics of constant curvature 1 with three conical singularities on the 2-sphere[J]. Illinois J Math, 2000, 44(1): 72-94.
    [6]
    EREMENKO A. Metrics of positive curvature with conical singularities on the sphere[J]. Proc Amer Math Soc, 2004, 132(11): 3349-3355.
    [7]
    FURUTA M, HATTORI Y. 2-dimensional singular spherical space forms[Z]. manuscript, 1998.
    [8]
    FUJIMORI S, KAWAKAMI Y, KOKUBU M, et al. CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere[J]. Proc Japan Acad Ser A Math Sci, 2011, 87(8): 144-149.
    [9]
    CHEN Q, WANG W, WU Y, et al. Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surface[J]. Pacific Journal of Mathematics, 2015, 273: 75-100.
    [10]
    BRYANT R L. Surfaces of mean curvature one in hyperbolic space[J]. Theorie des Varietes Minimales et Applications, 1988, 154-155: 321-347.
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