  In this work, we study the space of complete embedded rotationally symmetric
self-shrinking hypersurfaces in $\mathbb{R}^{n+1}$. First, using comparison
geometry in the context of metric geometry, we derive explicit upper bounds for
the entropy of all such self-shrinkers. Second, as an application we prove a
smooth compactness theorem on the space of all such shrinkers. We also prove
that there are only finitely many such self-shrinkers with an extra reflection
symmetry.
