Existence of Integral $m$-Varifolds minimizing $\int |A|^p$ and $\int |H|^p$, $p>m$, in Riemannian Manifolds

Details Existence of Integral $m$-Varifolds minimizing $\int |A|^p$ and $\int |H|^p$, $p>m$, in Riemannian Manifolds Existence of Integral $m$-Varifolds minimizing $\int |A|^p$ and $\int |H|^p$, $p>m$, in Riemannian Manifolds

2010-10-21

arxiv.org/abs/1010.4514v1

Mathematics

Differential Geometry

33 pages

Scientific paper

We prove existence and partial regularity 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 $(N,g)$, $2\leq mm$, 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.

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