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The second variation of nonorientable minimal submanifolds

Authors:	Marty Ross

Institution:	Department of Mathematics, Melbourne University, Parkville, Victoria, 3052, Australia

Abstract:	Suppose is a complete nonorientable minimal submanifold of a Riemannian manifold . We derive a second variation formula for the area of with respect to certain perturbations, giving a sufficient condition for the instability of . 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.

Keywords:	Nonorientable minimal surface stable Bernstein Theorem second variation

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