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Invariant CMC surfaces in \(\mathbb{H}^2\times\mathbb{R}\). (English) Zbl 1055.53045

Roughly speaking, in a 3-dimensional Riemannian manifold \(M^3\) the determination of all surfaces \(F^2\subset M^3\) of constant Gaussian curvature \(k\) or constant mean curvature \(h\), that are invariant under a 1-parametric group \(G\) of isometries of \(M^3\), leads to an ordinary second order differential equation. Namely, considering a local parametrization of \(F^2\) of the form \(X(s,t)=\text{Exp}(tX)\gamma(s)\), where \(X\) is an infinitesimal generator of \(G\) and \(\gamma(s)\) a parametrized curve in \(M\), by the invariance condition, \(k\) and \(h\) become functions of \(s\) alone.
In this paper, two types of subgroups \(G\subset\text{Isom}({\mathbb H}^2\times {\mathbb R})\), namely translations in direction of the \({\mathbb R}\)-factor and screw motions around the \({\mathbb R}\)-axis are considered. The corresponding surfaces of constant mean curvature are listed and their properties are discussed.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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