A Generalization of a Theorem of Delaunay on Constant Mean Curvature Surfaces

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Abstract

In 1841, Delaunay [2] classified all surfaces of revolution of constant mean curvature, with a beautifully simple description in terms of conics. The constancy of the mean curvature is expressed by an ordinary differential equation for the radius of rotation with respect to the meridian length. The resulting equation was, in those days, very familiar as the differential equation governing the roulette of a conic, that is, the locus of the focus of a conic as it rolls in a plane along a line.

Research supported by N.S.F. Grant D.M.S. 87–02383