On Surfaces that are Intrinsically Surfaces of Revolution

2016/07/28 by Daniel Freese, Freese, Daniel, Matthias Weber +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1607.08503

23 pages, 10 figures

arxiv created 2016/07/28 · arxiv updated 2016/07/29

Abstract

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.

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