Minimal isometric immersions into S^2 x R and H^2 x R Benoit Daniel 53C4253A1053C24isometric immersionminimal surfacehomogeneous Riemannian manifoldassociate familyrigidity For a given simply connected Riemannian surface $\Sigma$, we relate the problem of finding minimal isometric immersions of $\Sigma$ into $\mathbb{S}^2\times\mathbb{R}$ or $\mathbb{H}^2\times\mathbb{R}$ to a system of two partial differential equations on $\Sigma$. We prove that a constant intrinsic curvature minimal surface in $\mathbb{S}^2\times\mathbb{R}$ or $\mathbb{H}^2\times\mathbb{R}$ is either totally geodesic or part of an associate surface of a certain limit of catenoids in $\mathbb{H}^2\times\mathbb{R}$. We also prove that if a non-constant curvature Riemannian surface admits a continuous one-parameter family of minimal isometric immersions into $\mathbb{S}^2\times\mathbb{R}$ or $\mathbb{H}^2\times\mathbb{R}$, then all these immersions are associate. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5643 10.1512/iumj.2015.64.5643 en Indiana Univ. Math. J. 64 (2015) 1425 - 1445 state-of-the-art mathematics http://iumj.org/access/