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
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10.1512/iumj.2015.64.5643
10.1512/iumj.2015.64.5643
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Indiana Univ. Math. J. 64 (2015) 1425 - 1445
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