# Multiple valued Jacobi fields

## TL;DR

This paper introduces a multivalued theory for Jacobi fields on minimal submanifolds, defining and analyzing the existence, regularity, and superposition properties of Q-valued Jacobi fields in a Riemannian setting.

## Contribution

It develops a novel multivalued framework for Jacobi fields, extending classical theory to multiple valued sections and analyzing their structure and regularity.

## Key findings

- Q-valued Jacobi fields can be decomposed into classical Jacobi fields outside a small singular set.
- Existence and regularity results for minimizers of the second variation functional in the multivalued setting.
- The superposition principle holds for Q-valued Jacobi fields, except on a codimension at least two set.

## 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.