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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This work was partially supported by a grant from the Simons Foundation (246039 to Matthias Weber) and by a grant from the NSF (1461061 to Daniel Freese).
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Freese, D., Weber, M. On surfaces that are intrinsically surfaces of revolution. J. Geom. 108, 743–762 (2017). https://doi.org/10.1007/s00022-017-0370-6
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DOI: https://doi.org/10.1007/s00022-017-0370-6