Calculus of Variations and Geometric Measure Theory
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S. Stuvard

Multiple valued Jacobi fields

created by stuvard on 31 Jan 2017
modified on 08 Aug 2017

[BibTeX]

Preprint

Inserted: 31 jan 2017
Last Updated: 8 aug 2017

Pages: 72
Year: 2017

ArXiv: 1701.08753 PDF
Links: arXiv e-print

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.

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


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