  For hypersurfaces moving by standard mean curvature flow with boundary, we
show that if a tangent flow at a boundary singularity is given by a smoothly
embedded shrinker, then the shrinker must be non-orientable. We also show that
there is an initially smooth surface in 3-space that develops a boundary
singularity for which the shrinker is smoothly embedded (and therefore
non-orientable). Indeed, we show that there is a nonempty open set of such
initial surfaces.
  Let k be the largest number with the following property: if M is a minimal
surface in 3-space bounded by a smooth simple closed curve of total curvature
less than k, then M is a disk. Examples show that $k<4\pi$. In this paper, we
use mean curvature flow to show that $k >3\pi$. We get a slightly larger lower
bound for orientable surfaces.
