Asymptotic structure of self-shrinkers

TL;DR

This paper characterizes the asymptotic geometry of noncompact self-shrinkers in three-dimensional space, showing they resemble cones or cylinders at infinity, which advances understanding of their structure in geometric analysis.

Contribution

It establishes the asymptotic behavior of ends of noncompact self-shrinkers with finite topology, revealing they are smoothly asymptotic to cones or cylinders, a new insight in mean curvature flow.

Findings

Ends are asymptotic to cones or cylinders

Provides a classification of asymptotic structures

Enhances understanding of self-shrinker geometry

Abstract

We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.