Multiple valued Jacobi fields

Abstract

We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold $\Sigma$ of a Riemannian manifold $\mathcal{M}$. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of multiple valued sections of the normal bundle of $\Sigma$ in $\mathcal{M}$, and we study existence and regularity of such minimizers. Finally, we prove that any $Q$-valued Jacobi field can be written as the superposition of $Q$ classical Jacobi fields everywhere except for a relatively closed singular set having codimension at least two in the domain.