Areas of spherical polyhedral surfaces with regular faces
Y Akama, B Hua, Y Su - arXiv preprint arXiv:1804.11033, 2018 - arxiv.org
Y Akama, B Hua, Y Su
arXiv preprint arXiv:1804.11033, 2018•arxiv.orgFor 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, ie the total angle at each vertex is at most $2\pi $. We classify all spherical
tilings with regular spherical polygons, ie total angles at vertices are exactly $2\pi $. We …
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, ie the total angle at each vertex is at most $2\pi $. We classify all spherical
tilings with regular spherical polygons, ie total angles at vertices are exactly $2\pi $. We …
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 . We classify all spherical tilings with regular spherical polygons, i.e. total angles at vertices are exactly . 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 for some . That is, we obtain a definite gap between the area of such a surface and that of the unit sphere.
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