Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

Abstract

Given a smooth, symmetric, homogeneous of degree one function $f\left({\lambda }_1,\cdots ,{\lambda }_n\right)$ satisfying ${\partial }_{i}f>0$ for all $i=1,\cdots ,n$, and a rotationally symmetric cone $\mathcal{C}$ in ${\mathbb{R}}^{n+1}$, we show that there is a $f$ self-shrinker (i.e. a hypersurface $\Sigma$ in ${\mathbb{R}}^{n+1}$ which satisfies $f\left({\kappa }_1,\cdots ,{\kappa }_n\right)+\frac{1}{2}X\cdot N=0$, 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 $\mathcal{C}$ at infinity.

arXiv commentarXiv:1604.08577v1 was split into arXiv:1604.08577v2 and this paper; published in Journal of Differential Equations