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 $M$, 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.
Multiple valued Jacobi fields / S. Stuvard. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 58:3(2019 May 08), pp. 92.1-92.83.
Multiple valued Jacobi fields
S. Stuvard
2019
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 $M$, 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.File | Dimensione | Formato | |
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Stuvard2019_Article_MultipleValuedJacobiFields.pdf
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