Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C45, 58E12

Retrieve articles in all journals with MSC (1991): 53C45, 58E12

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

Keywords: Nonorientable minimal surface, stable, Bernstein Theorem, second variation
Received by editor(s): July 21, 1994
Article copyright: © Copyright 1997 American Mathematical Society