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Parabolic and hyperbolic screw motion surfaces in \(\mathbb H^{2}\times \mathbb R\). (English) Zbl 1178.53060
The author studies certain equivariant minimal and constant mean curvature surfaces in the Riemannian product space \(H^2\times \mathbb R\).
More precisely, he determines families of complete minimal and constant mean curvature surfaces in \(H^2\times \mathbb R\), where \(H^2\) is the hyperbolic half plane with metric \({dx^2+dy^2\over y^2}\), of the form \((x,y)\mapsto(x,y,\ell x+\lambda(y))\) (“parabolic screw motion surface”) or \((\varphi,\rho)\mapsto(e^\varphi\cos\rho,e^\varphi\sin\rho, \ell\varphi+\lambda(\rho))\), \(\rho\in(0,\pi)\), (“hyperbolic screw motion surface”). Explicit formulae are obtained in each case.
Conjugate and associate minimal surfaces are then studied. The author shows that isometric minimal parabolic screw motion surfaces are associate. Conjugacy is discussed for several cases.

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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