  Using a maximum principle for self-shrinkers of the mean curvature flow, we
give new proofs of a rigidity theorem for rotationally symmetric compact
self-shrinkers and a result about the asymptotic behavior of self-shrinkers.
This comparison argument also implies a linear bound for the second fundamental
form of self-shrinking surfaces under natural assumptions. As a consequence,
translating solitons can be related to these self-shrinkers.
