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The second variation of nonorientable minimal submanifolds

Author(s): Marty Ross
Journal: Trans. Amer. Math. Soc. 349 (1997), 3093-3104.
MSC (1991): Primary 53C45; Secondary 58E12
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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: 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
Copyright of article: Copyright 1997, American Mathematical Society

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