What's the defination of homothetically expanding soliton of mean curvature flow? Thanks!
I think it just means that there is a hypersurface $\Sigma_t\subset \mathbb{R}^n$ satisfying meancurvature flow, such that for every $\Sigma_t$, there is a homothety $\varphi_t: \mathbb{R}^n\to \mathbb{R}^n$ such that $\varphi_t(\Sigma_t)=\Sigma_0$. A homothety $\varphi$ is a map of the form $x\mapsto Ax+b$, where $b\in\mathbb{R}^n$, and $A$ is in the conformal group, so that there exists $\lambda\in \mathbb{R}^\times$ such that $\lambda\cdot A \in O(n)$ (or it might be required that $\lambda\cdot A=I$, depending on the interpretation of homothety). 

