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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds


Author: John Douglas Moore
Journal: Trans. Amer. Math. Soc. 358 (2006), 5193-5256
MSC (2000): Primary 53C40, 58E12; Secondary 58D15, 58E05
Published electronically: July 21, 2006
Erratum: Tran. Amer. Math. Soc. 359 (2007) 5117-5123.
MathSciNet review: 2238914
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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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Additional Information

John Douglas Moore
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: moore@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04317-0
PII: S 0002-9947(06)04317-0
Received by editor(s): February 18, 2004
Published electronically: July 21, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.