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PERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS

Part of: Global differential geometry

Published online by Cambridge University Press:  15 October 2013

ANA MENEZES
ANA MENEZES*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro-RJ, Brazil email anamaria@impa.br
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Abstract

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In this paper we prove the existence of complete minimal surfaces in some metric semidirect products. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in ${ \mathbb{R} }^{3} $. In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in ${\mathrm{Sol} }_{3} $ with a one-parameter family of nonisometric metrics.

Keywords

semidirect productsminimal surfaces

MSC classification

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 1 , February 2014 , pp. 127 - 144
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Jost, J., ‘Conformal mappings and the Plateau–Douglas problem in Riemannian manifolds’, J. reine angew. Math. 359 (1985), 37–54.Google Scholar
Mazet, L., Rodríguez, M. and Rosenberg, H., ‘Periodic constant mean curvature surfaces in ${ \mathbb{H} }^{2} \times \mathbb{R} $’, Asian J. Math., to appear.Google Scholar
Meeks, W. H. III, Mira, P., Pérez, J. and Ros, A., ‘Constant mean curvature spheres in homogeneous three-manifolds’, in preparation.Google Scholar
Meeks, W. H. III and Pérez, J., ‘Constant mean curvature surfaces in metric Lie groups’, in: Geometric Analysis: Partial Differential Equations and Surfaces, Contemporary Mathematics, 570 (American Mathematical Society, Providence, RI, 2012), 25–110.Google Scholar
Rosenberg, H., ‘Minimal surfaces in ${ \mathbb{M} }^{2} \times \mathbb{R} $’, Illinois J. Math. 46 (4) (2002), 1177–1195.Google Scholar
Rosenberg, H., Souam, R. and Toubiana, E., ‘General curvature estimates for stable $H$-surfaces in 3-manifolds and applications’, J. Differential Geom. 84 (3) (2010), 623–648.CrossRefGoogle Scholar