  This paper concerns integral varifolds of arbitrary dimension in an open
subset of Euclidean space satisfying integrability conditions on their first
variation. Firstly, the study of pointwise power decay rates almost everywhere
of the quadratic tilt-excess is completed by establishing the precise decay
rate for two-dimensional integral varifolds of locally bounded first variation.
In order to obtain the exact decay rate, a coercive estimate involving a
height-excess quantity measured in Orlicz spaces is established. Moreover,
counter-examples to pointwise power decay rates almost everywhere of the
super-quadratic tilt-excess are obtained. These examples are optimal in terms
of the dimension of the varifold and the exponent of the integrability
condition in most cases, for example if the varifold is not two-dimensional.
These examples also demonstrate that within the scale of Lebesgue spaces no
local higher integrability of the second fundamental form, of an at least
two-dimensional curvature varifold, may be deduced from boundedness of its
generalised mean curvature vector.
  Amongst the tools are Cartesian products of curvature varifolds.
