Construction of Surfaces with Constant Mean Curvature
Abstract
The paper studies surfaces with constant mean curvature (CMC) H. If H = 0 then the surfaces are minimal. CMC tori were studied by H. Wente. U. Abresz proved that Wente tori have one family of planar lines of curvature and characterized them with elliptic integrals.
A.I. Bobenko in his studies considered the problem of constructing CMC tori E3, S3, H3. In this paper, CMC surfaces of revolution are investigated. For a surface in E3 the Bonnet’s theorem states that for any surface having constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature.
According to this statement, for surfaces of revolution with constant positive Gaussian curvature, CMC surfaces are constructed using the Bonnet’s theorem. It is proved that constructed surfaces are also surfaces of revolution. A family of plane curvature lines (meridians) is described by elliptic integrals, and surfaces with Gaussian curvature are also described by elliptic integrals. These surfaces are constructed using the mathematical software package.
DOI 10.14258/izvasu(2018)4-22
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Copyright (c) 2018 М.А. Чешкова, И.В. Поликанова
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