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

TL;DR

This paper proves the uniqueness of certain self-shrinking hypersurfaces under degree-one curvature flow that are asymptotic to a given cone at infinity, under specific conditions on the flow function and cone geometry.

Contribution

It establishes the first uniqueness result for self-shrinkers with prescribed asymptotic cone under a broad class of curvature flows.

Findings

At most one self-shrinker asymptotic to a given cone exists under specified conditions.

Conditions on the curvature flow function and cone derivatives are crucial for the uniqueness.

The result applies to a class of homogeneous degree-one functions satisfying positivity of partial derivatives.

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.