Metrics of positive curvature with conic singularities on the sphere

By Alexandre Eremenko. 2002

113 citations recorded by the source corpus.

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers θ 1 , θ 2 , θ 3 for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles 2πθ 1 , 2πθ 2 , 2πθ 3 . The same condition is necessary and sufficient for the triple πθ 1 , πθ 2 , πθ 3 to be interior angles of a spherical triangular membrane.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7ebb2577391f79b812f70b4db9006a4c675729c3

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Metrics with conic singularities and spherical polygons (2014, background): “In [36], [4], all possibilities for spherical triangles are completely described, see also [10] where a minor error in [4, Theorem 2] is corrected.”
Ricci Flow on Surfaces with Conical Singularities (2009, background): “See the discussion in [12], [10] and [11].”
On the moduli space of singular Euclidean surfaces (2007, background): “The case of spherical metric is more delicate, see [12, 35] for a study of spherical metric with three conical singularities on the 2-sphere.”
Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces (2013, background): “The case where D is an effective Z-divisor supported at two or three points was proved in [Troyanov 1989; Furuta and Hattori 1998; Umehara and Yamada 2000; Eremenko 2004].”
Mean Field Equation of Liouville Type with Singular Data: Topological Degree (2015, background): “This problem has been completely solved if the number of singularities is less than four; see [27, 55].”