Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry

Abstract In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 {\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.