Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends

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 embedded self-shrinking ends in ${\mathbb{R}}^{n+1}$ that are asymptotic to $C$. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete properly embedded self-shrinker asymptotic to it.