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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Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal surfaces. I, volume 295 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1992). Boundary value problems.
do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice Hall, New Jersey (1976)
Enneper, A.: Weitere bemerkungen über asymptotische linien. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, pp. 2–23 (1871)
Karcher, H.: Construction of minimal surfaces. Surveys in Geometry, pp. 1–96, 1989. University of Tokyo, 1989, and Lecture Notes No. 12, SFB256, Bonn (1989)
Smyth, B.: A Generalization of a Theorem of Delaunay on Constant Mean Curvature Surfaces, pp. 123–130. Springer, New York (1993)
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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