Compactness and finiteness theorems for rotationally symmetric self shrinkers

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.