  Self-shrinkers are hypersurfaces that shrink homothetically under mean
curvature flow; these solitons model the singularities of the flow. It it
presently known that an entire self-shrinking graph must be a hyperplane. In
this paper we show that the hyperplane is rigid in an even stronger sense,
namely: For $2 \leq n \leq 6$, any smooth, complete self-shrinker
$\Sigma^n\subset\mathbf{R}^{n+1}$ that is graphical inside a large, but
compact, set must be a hyperplane. In fact, this rigidity holds within a larger
class of almost stable self-shrinkers.
  A key component of this paper is the procurement of linear curvature
estimates for almost stable shrinkers, and it is this step that is responsible
for the restriction on $n$. Our methods also yield uniform curvature bounds for
translating solitons of the mean curvature flow.
