A Remark on Soliton Equation of Mean Curvature Flow

Abstract

In this short note, we consider self-similar immersions $F:{\mathbb{R}}^n\to {\mathbb{R}}^{n+k}$ of the Graphic Mean Curvature Flow of higher co-dimension. We show that the following is true: Let $F(x)=(x,f(x)),x\in {\mathbb{R}}^n$ be a graph solution to the soliton equation $$\bar{H}(x)+F^{\perp }(x)=0.$$ Assume ${\sup }_{{\mathbb{R}}^n}|Df(x)|\le C_0<+\infty$. Then there exists a unique smooth function $f_{\infty }:{\mathbb{R}}^n\to {\mathbb{R}}^k$ such that $$f_{\infty }(x)=\underset{\hat{I}»\to \infty }{\lim }{f}_{\hat{I}}»(x)$$ and $$f_{\infty }(rx)=r{f}_{\infty }(x)$$ for any real number $r\ne 0$, where $$f_{\hat{I}}»(x)=\hat{I}{»}^{-1}f(\hat{I}»x).$$

6 pages