  In this note we first show a compactness theorem for rotationally symmetric
self shrinkers of entropy less than 2, concluding that there are entropy
minimizing self shrinkers diffeomorphic to $S^1 \times S^{n-1}$ for each $n
\geq 2$ in the class of rotationally symmetric self shrinkers. Assuming extra
symmetry, namely that the profile curve is convex, we remove the entropy
assumption. Supposing the profile curve is additionally reflection symmetric we
show there are only finitely many such shrinkers up to rigid motion.
