Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-18T12:29:13.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Helicoidal minimal surfaces in hyperbolic space

Published online by Cambridge University Press:  22 January 2016

Jaime B. Ripoll
Jaime B. Ripoll*
Affiliation:
Universidade Federal do Rio Grande do Sul, Instituto de Matemática, Av. Bento Gonçalves, 9500, 91. 500-Porto Alegre-RS, Brazil
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Denote by H3 the 3-dimensional hyperbolic space with sectional curvatures equal to – 1, and let g be a geodesic in H3 Let {ψt} be the translation along g (see § 2) and let {φt} be the one-parameter subgroup of isometries of H3 whose orbits are circles centered on g. Given any α ∊ R, one can show that λ = {λt} = ψt ∘ φαt} is a one-parameter subgroup of isometries of H3 (see § 2) which is called a helicoidal group of isometries with angular pitch α. Any surface in H3 which is λ-invariant is called a helicoidal surface.

In this work we prove some results concerning minimal helicoidal surfaces in H3. The first one reads:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 114 , June 1989 , pp. 65 - 75
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

Anderson, M. T., Complete minimal varieties in hyperbolic space, Invent. Math., 69 (1982), 477–494.Google Scholar
do Carmo, M. P. Gomes, J. de M., Thorbergsson, G., The influence of the boundary behaviour on hypersurf aces with constant mean curvature in Hn+1 , Comment. Math. Helv., 61 (1986), 429–441.Google Scholar
do Carmo, M. P. Lawson, H. B., On Alexandrov-Bernstein theorems in hyperbolic space, Duke Math. J., 50 (1984), 995–1003.Google Scholar
Gomes, J. de M., Ripoll, J. B., Rodriguez, L., On surfaces of constant mean curvature in hyperbolic space, preprint (IMPA), 1985.Google Scholar
Levitt, G., Rosenberg, H., Symmetry of constant mean curvature hypersurf aces in hyperbolic space, Duke Math. J., 52, no. 1 (1985).Google Scholar
Ripoll, J. B., Superfícies invariantes de curvatura média constante, Tese de Doutorado, IMPA, 1986.Google Scholar