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PARABOLIC AND HYPERBOLIC SCREW MOTION SURFACES IN ℍ2×ℝ

Part of: Global differential geometry

Published online by Cambridge University Press:  01 August 2008

RICARDO SA EARP
RICARDO SA EARP*
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, 22453-900 RJ, Brazil (email: [email protected])
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Abstract

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In thispaper we find many families in the product space ℍ2×ℝ of complete embedded, simply connected, minimal and surfaces with constant mean curvature H such that |H|≤1/2. We study complete surfaces invariant either by parabolic or by hyperbolic screw motions. We study the notion of isometric associate immersions. We exhibit an explicit formula for a Scherk-type minimal surface. We give a one-parameter family of entire vertical graphs of mean curvature 1/2. We prove a generalized Bour lemma that can be applied to ℍ2×ℝ,𝕊2×ℝ and to Heisenberg’s space to produce a family of screw motion surfaces isometric to a given one.

Keywords

minimal and constant mean curvature surfacesisometricassociateembeddedvertical and horizontal graphs

MSC classification

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 1 , August 2008 , pp. 113 - 143
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Abresch, U. and Rosenberg, H., ‘A Hopf differential for constant mean curvature surfaces in S 2×R and H 2×R’, Acta Math. 193(2) (2004), 141–174.Google Scholar
[2]Daniel, B., ‘Isometric immersions into 𝕊n×ℝ and ℍn×ℝ and applications to minimal surfaces’, in: Prépublications, Vol. 375 (Institut de Math. de Jussieu, 2004).Google Scholar
[3]Bour, E., ‘Mémoire sur la déformation des surfaces’, J. Ecole Pol. 39 (1862), 1–148.Google Scholar
[4]Do Carmo, M. P. and Dajczer, M., ‘Helicoidal surfaces with constant mean curvature’, Tohoku Math. J. 34 (1982), 425–435.Google Scholar
[5]Collin, P. and Rosenberg, H., ‘Construction of harmonic diffeomorfisms and minimal graphs’, Preprint (arXiv: math.DG/0701547v1).Google Scholar
[6]Dierkes, U., Hildebrandt, S., Küster, A. and Wohlrab, O., Minimal Surfaces, Vol. I: Boundary Value Problems and Vol. II Boundary Regularity (Grundlehren Der Mathematischen Wissenschaften) 295 and 296 (Springer, Berlin, 1992).Google Scholar
[7]Fernández, I. and Mira, P., ‘Harmonic maps and constant mean curvature surfaces in ℍ2×ℝ,’, Amer. J. Math. 129(4) (2007), 1145–1181.CrossRefGoogle Scholar
[8]Fernández, I. and Mira, P., ‘A characterization of constant mean curvature surfaces in homogeneous 3-manifold,’, Differential Geom. Appl. 25(3) (2007), 281–289.Google Scholar
[9]Fernández, I. and Mira, P., ‘Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space’, Preprint.Google Scholar
[10]Hauswirth, L., ‘Generalized Riemann examples in three-dimensional manifolds’, Pacific J. Math. 224(1) (2006), 91–117.Google Scholar
[11]Hauswirth, L., Sa Earp, R. and Toubiana, E., ‘Associate and conjugate minimal immersions in M 2×R’, Tohoku Math. J. 60 (2008), 267–286.Google Scholar
[12]Hauswirth, L., Rosenberg, H. and Spruck, J., ‘Infinite boundary value problems for constant mean curvature graphs in ℍ2×ℝ and 𝕊2×ℝ’, http://www.math.jhu.edu/∼js/jsfinal.pdf.Google Scholar
[13]Meeks III, W. and Rosenberg, H., ‘The theory of minimal surfaces in M×ℝ’, Comment. Math. Helv. 80(4) (2005), 811–858.Google Scholar
[14]Nelli, B. and Rosenberg, H., ‘Minimal surfaces in ℍ2×ℝ’, Bull. Braz. Math. Soc. 33 (2002), 263–292.CrossRefGoogle Scholar
[15]Nelli, B. and Rosenberg, H., ‘Global properties of constant mean curvature surfaces in ℍ2×ℝ’, Pacific J. Math. 226(1) (2006), 137–152.Google Scholar
[16]Ordóñes, J., ‘Superfícies helicoidais com curvatura constante no espaço de formas tridimensionais’, Doctoral Thesis, PUC-Rio, 1995.Google Scholar
[17]Rosenberg, H., ‘Minimal surfaces in 𝕄2×ℝ’, Illinois J. Math. 46 (2002), 1177–1195.Google Scholar
[18]Sa Earp, R. and Toubiana, E., ‘Existence and uniqueness of minimal graphs in hyperbolic space’, Asian J. Math. 4 (2000), 669–694.CrossRefGoogle Scholar
[19]Sa Earp, R. and Toubiana, E., ‘Screw motion surfaces in ℍ2×ℝ and 𝕊2×ℝ’, Illinois J. Math. 49(3) (2005), 1323–1362.Google Scholar
[20]Schoen, R. and Yau, S. T., ‘Lectures on harmonic maps’, Conference Proceedings and Lecture Notes in Geometry and Topology, II (International Press, Cambridge, MA, 1997).Google Scholar
[21]Spruck, J., ‘Interior gradient estimates and existence theorems for constant mean curvature graphs in M n×ℝ’, Pure Appl. Math. Q. 3(3) (2007), 785–800.CrossRefGoogle Scholar
[22]Wan, T. Y. H., ‘Constant mean curvature surface, harmonic maps, and universal Teichmüller space’, J. Differential Geom. 35 (1992), 643–657.Google Scholar