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- W2798409823 abstract "For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of planar graphs which admit spherical polyhedral surfaces with the curvature bounded below by 1 in the sense of Alexandrov, i.e. the total angle at each vertex is at most $2pi$. We classify all spherical tilings with regular spherical polygons, i.e. total angles at vertices are exactly $2pi$. We prove that for any graph in this class which does not admit a spherical tiling, the area of the associated spherical polyhedral surface with the curvature bounded below by 1 is at most $4pi - epsilon_0$ for some $epsilon_0 > 0$. That is, we obtain a definite gap between the area of such a surface and that of the unit sphere." @default.
- W2798409823 created "2018-05-07" @default.
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- W2798409823 date "2018-04-29" @default.
- W2798409823 modified "2023-09-24" @default.
- W2798409823 title "Areas of spherical polyhedral surfaces with regular faces" @default.
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- W2798409823 doi "https://doi.org/10.48550/arxiv.1804.11033" @default.
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