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Existence of integral m-varifolds minimizing $\int |A|^p $ and $\int |H|^p$ , p>m, in Riemannian manifolds

TitleExistence of integral m-varifolds minimizing $\int |A|^p $ and $\int |H|^p$ , p>m, in Riemannian manifolds
Publication TypeJournal Article
Year of Publication2014
AuthorsMondino, A
JournalCalculus of Variations and Partial Differential Equations
Volume49
Pagination431–470
Date PublishedJan
ISSN1432-0835
Abstract

We prove existence of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2 \leq m<n$ and $p>m$ under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in ${\mathbb{R }^S}$ involving $\int  |H|^p$to  avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.

URLhttps://doi.org/10.1007/s00526-012-0588-y
DOI10.1007/s00526-012-0588-y

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