Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds

Moore, John

doi:10.1090/S0002-9947-06-04317-0

Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds
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Trans. Amer. Math. Soc. 358 (2006), 5193-5256 Request permission

Erratum: Trans. Amer. Math. Soc. 359 (2007), 5117-5123.

Abstract:

The purpose of this article is to study conformal harmonic maps $f:\Sigma \rightarrow M$, where $\Sigma$ is a closed Riemann surface and $M$ is a compact Riemannian manifold of dimension at least four. Such maps define parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold $M$ is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of automorphisms of $\Sigma$. They are Morse nondegenerate in the usual sense if $\Sigma$ has genus at least two, lie on two-dimensional nondegenerate critical submanifolds if $\Sigma$ has genus one, and on six-dimensional nondegenerate critical submanifolds if $\Sigma$ has genus zero.

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