Ribaucour transformations for flat surfaces in the hyperbolic 3-space. (English) Zbl 1308.53062
Summary: We consider Ribaucour transformations for flat surfaces in the hyperbolic 3-space, \(\mathbb H^3\). We show that such transformations produce complete, embedded ends of horosphere type and curves of singularities which generically are cuspidal edges. Moreover, we prove that these ends and curves of singularities do not intersect. We apply Ribaucour transformations to rotational flat surfaces in \(\mathbb H^3\) providing new families of explicitly given flat surfaces \(\mathbb H^3\) which are determined by several parameters. For special choices of the parameters, we get surfaces that are periodic in one variable and surfaces with any even number or an infinite number of embedded ends of horosphere type.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
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