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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 \(f_ t\).

MSC:

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

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