  In this paper, we introduce a version of the moving plane method that applies
to potentially quite singular hypersurfaces, generalizing the classical moving
plane method for smooth hypersurfaces. Loosely speaking, our version for
varifolds shows that smoothness and symmetry at infinity (respectively at the
boundary) can be promoted to smoothness and symmetry in the interior. The key
feature, in contrast with the classical formulation of the moving plane
principle, is that smoothness is a conclusion rather than an assumption.
  We implement our moving plane method in the setting of compact varifolds with
smooth boundary and in the setting of complete varifolds. A key ingredient is a
Hopf lemma for stationary and CMC-varifolds. Our Hopf lemma provides a new tool
to establish smoothness of varifolds, and works in arbitrary dimensions and
without any stability assumptions. As applications of our new moving plane
method, we prove varifold uniqueness results for the catenoid, spherical caps,
and Delaunay surfaces that are inspired by classical uniqueness results by
Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold
version of Alexandrov's Theorem for compact CMC-varifolds in hyperbolic space.
