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.

}, issn = {1432-0835}, doi = {10.1007/s00526-012-0588-y}, url = {https://doi.org/10.1007/s00526-012-0588-y}, author = {Andrea Mondino} }