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

 

The second variation of
nonorientable minimal submanifolds


Author: Marty Ross
Journal: Trans. Amer. Math. Soc. 349 (1997), 3093-3104
MSC (1991): Primary 53C45; Secondary 58E12
MathSciNet review: 1422909
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Abstract: Suppose $M$ is a complete nonorientable minimal submanifold of a Riemannian manifold $N$. We derive a second variation formula for the area of $M$ with respect to certain perturbations, giving a sufficient condition for the instability of $M$. Some simple applications are given: we show that the totally geodesic $\mathbb {R} \mathbb {P}^{2}$ is the only stable surface in $\mathbb {R} \mathbb {P}^{3}$, and we show the non-existence of stable nonorientable cones in $\mathbb {R}^{4}$. We reproduce and marginally extend some known results in the truly non-compact setting.


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Additional Information

Marty Ross
Affiliation: Department of Mathematics, Melbourne University, Parkville, Victoria, 3052, Australia
Address at time of publication: Antarctic CRC, Box 252-80, Hobart, Tasmania, Australia
Email: marty@mundoe.maths.mu.oz.au

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01936-3
PII: S 0002-9947(97)01936-3
Keywords: Nonorientable minimal surface, stable, Bernstein Theorem, second variation
Received by editor(s): July 21, 1994
Article copyright: © Copyright 1997 American Mathematical Society