Multiple valued Jacobi fields

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.