Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Abstract

Given a smooth, symmetric, homogeneous of degree one function $f=f\left({\lambda }_1,\cdots ,{\lambda }_n\right)$ satisfying ${\partial }_{i}f>0$ for all $i=1,\cdots ,n$, and an oriented, properly embedded smooth cone ${\mathcal{C}}^n$ in ${\mathbb{R}}^{n+1}$, we show that under some suitable conditions on $f$ and the covariant derivatives of the second fundamental form of $\mathcal{C}$, there is at most one $f$ self-shrinker (i.e. an oriented hypersurface ${\Sigma }^n$ in ${\mathbb{R}}^{n+1}$ for which $f\left({\kappa }_1,\cdots ,{\kappa }_n\right)+\frac{1}{2}X\cdot N=0$ holds, where $X$ is the position vector, $N$ is the unit normal vector, and ${\kappa }_1,\cdots ,{\kappa }_n$ are principal curvatures of $\Sigma$) that is asymptotic to the given cone $\mathcal{C}$ at infinity.