Associate and conjugate minimal immersions in M x R

We establish the definition of associate and conjugate conformal minimal isometric immersions into the product spaces, where the first factor is it 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 it 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 it 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.