A barrier principle at infinity for varifolds with bounded mean curvature

Our work investigates varifolds in a Riemannian manifold, with arbitrary codimension and with bounded mean curvature, which are contained in an open domain. Under mild assumptions on the curvatures of the ambient manifold and on the boundary of the domain, also allowing for certain singularities, we prove a barrier principle at infinity, namely we show that the distance between the submanifold and the boundary of the domain equals the distance between the boundary of the submanifold and the boundary of the domain. Our theorem is a consequence of sharp maximum principles at innity on varifolds, of independent interest.