Calculus of Variations and Geometric Measure Theory

S. Stuvard

Multiple valued Jacobi fields

created by stuvard on 31 Jan 2017
modified on 27 Jun 2019


Published Paper

Inserted: 31 jan 2017
Last Updated: 27 jun 2019

Journal: Calc. Var. Partial Differential Equations
Volume: 58
Number: 3
Pages: 83
Year: 2019
Doi: 10.1007/s00526-019-1545-9

ArXiv: 1701.08753 PDF
Links: Webpage


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.

Keywords: Almgren's $Q$-valued functions, second variation, stability operator, regularity