Constant mean curvature planes with inner rotational symmetry in Euclidean 3-space. (English) Zbl 0799.53011

Let \(({\mathbb{R}}^ 2,ds^ 2)\) be a two-dimensional Riemannian manifold admitting an isometric \(S^ 1\)-action with a fixed point \(p\). The authors prove that for each \(m\in{\mathbb{N}}\) there exists exactly a 1- parameter family of conformal isometric immersions \(f_ t: {\mathbb{R}}^ 2\to{\mathbb{R}}^ 3\) into Euclidean 3-space where \(f_ t\) has constant mean curvature and \(p\) is an umbilic of order \(m\). The authors investigate the global behaviour of \(f_ t\).


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C10 \(G\)-structures
Full Text: DOI EuDML


[1] Baouendi, M.S., Goulaouic, C.: Singular Nonlinear Cauchy-Problems. J. Differ. Equations22, 268–291 (1976) · Zbl 0344.35012 · doi:10.1016/0022-0396(76)90028-0
[2] Pinkall, U., Sterling, I.: On the classification of constant mean curvature tori. Ann. Math.130, 407–451 (1989) · Zbl 0683.53053 · doi:10.2307/1971425
[3] Smyth, B.: The generalization of Delaunay’s Theorem to constant mean curvature surfaces with continuous internal symmetrie. (Preprint 1987)
[4] Timmreck, M.:H-Flächen mit einer einparametrigen Symmetrie. Diplomarbeit an der TU Berlin. Berlin 1990
[5] Bobenko, A.: Constant mean curvature surfaces and integrable equations. Russ. Math. Surv.46:4 (1991) · Zbl 0780.53009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.