The second variation of nonorientable minimal submanifolds

Ross, Marty

doi:10.1090/S0002-9947-97-01936-3

The second variation of nonorientable minimal submanifolds
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Trans. Amer. Math. Soc. 349 (1997), 3093-3104 Request permission

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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