A Remark on Soliton Equation of Mean Curvature Flow

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 $$\bar{H}(x)+F^{\mathrm{\perp }}(x)=0.$$ Assume ${\sup }_{{\mathbb{R}}^n}\mathrm{\mid }Df(x)\mathrm{\mid }\le C_0<+\mathrm{\infty }$. Then there exists a unique smooth function $f_{\mathrm{\infty }}:{\mathbb{R}}^n\to {\mathbb{R}}^k$ such that $$f_{\mathrm{\infty }}(x)=\underset{\lambda \to \mathrm{\infty }}{\lim }{f}_{\lambda }(x)$$ and $$f_{\mathrm{\infty }}(rx)=r{f}_{\mathrm{\infty }}(x)$$ for any real number $r≠0$, where $$f_{\lambda }(x)={\lambda }^{-1}f(\lambda x)\mathrm{.}$$