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2008 Associate and conjugate minimal immersions in $\boldsymbol{M} \times \boldsymbol{R}$
Laurent Hauswirth, Ricardo Sa Earp, Eric Toubiana
Tohoku Math. J. (2) 60(2): 267-286 (2008). DOI: 10.2748/tmj/1215442875

Abstract

We establish the definition of associate and conjugate conformal minimal isometric immersions into the product spaces, where the first factor is a Riemannian surface and the other is the set of real numbers. When the Gaussian curvature of the first factor is nonpositive, we prove that an associate surface of a minimal vertical graph over a convex domain is still a vertical graph. This generalizes a well-known result due to R. Krust. Focusing the case when the first factor is the hyperbolic plane, it is known that in certain class of surfaces, two minimal isometric immersions are associate. We show that this is not true in general. In the product ambient space, when the first factor is either the hyperbolic plane or the two-sphere, we prove that the conformal metric and the Hopf quadratic differential determine a simply connected minimal conformal immersion, up to an isometry of the ambient space. For these two product spaces, we derive the existence of the minimal associate family.

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Laurent Hauswirth. Ricardo Sa Earp. Eric Toubiana. "Associate and conjugate minimal immersions in $\boldsymbol{M} \times \boldsymbol{R}$." Tohoku Math. J. (2) 60 (2) 267 - 286, 2008. https://doi.org/10.2748/tmj/1215442875

Information

Published: 2008
First available in Project Euclid: 7 July 2008

zbMATH: 1153.53041
MathSciNet: MR2428864
Digital Object Identifier: 10.2748/tmj/1215442875

Subjects:
Primary: 53C42

Rights: Copyright © 2008 Tohoku University

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