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 $$ \bar{H}(x) + F^{\bot}(x) = 0. $$ 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 $$ f_{\infty}(x) = \lim_{\lambda \to \infty}f_{\lambda}(x) $$ and $$ f_{\infty}(r x)=r f_{\infty}(x) $$ for any real number $r\not= 0$, where $$ f_{\lambda}(x) = \lambda^{-1}f(\lambda x). $$