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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 $(N,g)$, $2\leqm<n$ and $p>m$, under suitable assumptions on $N$ (in the end of the paper wegive many examples of such ambient manifolds). To this aim we introduce thefollowing new tools: some monotonicity formulas for varifolds in $\mathbb{R}^S$involving $\int |H|^p$, to avoid degeneracy of the minimizer, and a sort ofisoperimetric inequality to bound the mass in terms of the mentionedfunctionals.

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