A barrier principle at infinity for varifolds with bounded mean curvature

Our work investigates varifolds $\Sigma \subset M$ in a Riemannian manifold, with arbitrary codimension and bounded mean curvature, contained in an open domain $\Omega$. Under mild assumptions on the curvatures of $M$ and on $\partial \Omega$, also allowing for certain singularities of $\partial \Omega$, we prove a barrier principle at infinity, namely we show that the distance of $\Sigma$ to $\partial \Omega$ is attained on $\partial \Sigma$. Our theorem is a consequence of sharp maximum principles at infinity on varifolds, of independent interest.