Poincar\'e and Sobolev type inequalities for intrinsic rectifiable varifolds

We prove a Poincar'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature bounded above. Our techniques allow us to consider Riemannian manifolds $(M^n,g)$ with $g$ of class $C^2$ or more regular, avoiding the use of Nash’s isometric embedding theorem. Our analysis permits to do some quite important fragments of geometric measure theory also for those Riemannian manifolds carrying a $C^2$ metric $g$, that is not $C^{k+\alpha }$ with $k+\alpha >2$. The class of varifolds we consider are those which first variation $\delta V$ lies in an appropriate Lebesgue space $L^p$ with respect to its weight measure $\mathrm{\parallel }V\mathrm{\parallel }$ with the exponent $p\in \mathbb{R}$ satisfying $p>k$.