  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.
