Metrics of positive curvature with conic singularities on the sphere

Eremenko, A.

doi:10.1090/S0002-9939-04-07439-8

Metrics of positive curvature with conic singularities on the sphere
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Proc. Amer. Math. Soc. 132 (2004), 3349-3355 Request permission

Abstract:

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers $\theta _1,\theta _2,\theta _3$ for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles $2\pi \theta _1,2\pi \theta _2, 2\pi \theta _3$. The same condition is necessary and sufficient for the triple $\pi \theta _1,\pi \theta _2,\pi \theta _3$ to be interior angles of a spherical triangular membrane.

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