## Tohoku Mathematical Journal

### Associate and conjugate minimal immersions in $\boldsymbol{M} \times \boldsymbol{R}$

#### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 60, Number 2 (2008), 267-286.

Dates
First available in Project Euclid: 7 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1215442875

Digital Object Identifier
doi:10.2748/tmj/1215442875

Mathematical Reviews number (MathSciNet)
MR2428864

Zentralblatt MATH identifier
1153.53041

#### Citation

Hauswirth, Laurent; Sa Earp, Ricardo; Toubiana, Eric. Associate and conjugate minimal immersions in $\boldsymbol{M} \times \boldsymbol{R}$. Tohoku Math. J. (2) 60 (2008), no. 2, 267--286. doi:10.2748/tmj/1215442875. https://projecteuclid.org/euclid.tmj/1215442875

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