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

We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int \mathrm{\mid }H{\mathrm{\mid }}^p$ and $\int \mathrm{\mid }A{\mathrm{\mid }}^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq m<n$and$p>m$,undersuitableassumptionson$N$ (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 $\int \mathrm{\mid }H{\mathrm{\mid }}^p$, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.