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Spherical metrics with conical singularities on 2-spheres

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Suppose that θ1,θ2,…,θn are positive numbers and n≥3. Does there exist a sphere with a spherical metric with n conical singularities of angles 2πθ1,2πθ2,…,2πθn? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994 We show that it is also necessary when we assume that θ1,θ2,…,θn∉N.

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