Entropy Bounds, Compactness and Finiteness Theorems for Embedded Self-shrinkers with Rotational Symmetry

TL;DR

This paper establishes entropy bounds, compactness, and finiteness results for complete embedded rotationally symmetric self-shrinkers in Euclidean space, advancing understanding of their geometric properties and classification.

Contribution

It provides explicit entropy bounds, a compactness theorem, and finiteness results for rotationally symmetric self-shrinkers, using comparison geometry techniques.

Findings

Explicit upper bounds for entropy of self-shrinkers

A smooth compactness theorem for the space of such shrinkers

Finiteness of self-shrinkers with reflection symmetry

Abstract

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.