Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends

摘要

Let $C\subset\mathbb{R}^{n+1}$ be a regular cone with vertex at the origin.In this paper, we show the uniqueness for smooth properly embeddedself-shrinking ends in $\mathbb{R}^{n+1}$ that are asymptotic to $C$. As anapplication, we prove that not every regular cone with vertex at the origin hasa smooth complete properly embedded self-shrinker asymptotic to it.