Browsing by Subject "surfaces of positive curvature"

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    Metrics with conic singularities and spherical polygons
    (2014) Eremenko, Alexandre; Gabrielov, Andrei; Tarasov, Vitaly; Department of Mathematics, School of Science
    A spherical nn-gon is a bordered surface homeomorphic to a closed disk, with nn distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 11, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are not multiples of ππ. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy.
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    Metrics with four conic singularities and spherical quadrilaterals
    (AMS, 2016) Eremenko, Alexandre; Gabrielov, Andrei; Tarasov, Vitaly; Department of Mathematical Sciences, School of Science
    A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that two angles at the corners are multiples of π. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy.
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    Spherical quadrilaterals with three non-integer angles
    (2016) Eremenko, Alexandre; Gabrielov, Andrei; Tarasov, Vitaly; Department of Mathematical Sciences, School of Science
    A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 11, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that one corner of a quadrilateral is integer (i.e., its angle is a multiple of ππ) while the angles at its other three corners are not multiples of ππ. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy, with the trivial monodromy at one of its four singular point.