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

TL;DR

This paper establishes Poincaré and Sobolev inequalities for functions on $k$-rectifiable varifolds within Riemannian manifolds, extending geometric measure theory techniques to less regular metrics without Nash's embedding.

Contribution

It introduces new inequalities for varifolds on Riemannian manifolds with $C^2$ metrics, avoiding Nash's theorem and broadening the scope of geometric measure theory.

Findings

Proved Poincaré and Sobolev inequalities for varifolds on Riemannian manifolds.

Extended geometric measure theory to manifolds with $C^2$ metrics.

Allowed analysis with varifolds whose first variation is in $L^p$, $p>k$.

Abstract

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 $\|V\|$ with the exponent $p\in\mathbb{R}$ satisfying $p>k$.