The second variation of nonorientable minimal submanifolds

Source abstract

Suppose M M is a complete nonorientable minimal submanifold of a Riemannian manifold N N . We derive a second variation formula for the area of M M with respect to certain perturbations, giving a sufficient condition for the instability of M M . Some simple applications are given: we show that the totally geodesic R P 2 \mathbb {R} \mathbb {P}^{2} is the only stable surface in R P 3 \mathbb {R} \mathbb {P}^{3} , and we show the non-existence of stable nonorientable cones in R 4 \mathbb {R}^{4} . We reproduce and marginally extend some known results in the truly non-compact setting.