Given an initial hypersurface in $\mathbb{R^{n+1}}$ with position vector $V_0$ there are two ways I have seen to define the evolution of $V_0$ due to a curvature flow;
$$(1) \hspace{1cm} \frac{\partial V}{\partial t}=\sigma(k_1,k_2,...,k_n)\hat{n}$$
$$\hspace{1cm}(2) \hspace{1cm} \left(\frac{\partial V}{\partial t}\right)^\perp=\sigma(k_1,k_2,...,k_n)\hat{n}$$
Where $V(x,t)$ satisfies one of the above equations and $V_0=V(x,0)$. Here $\sigma$ is a function of principal curvatures and $u^\perp=\langle u,\hat n \rangle \hat{n}$.
I have two questions concerning this.
- Do these equations provide the same family of hypersurfaces $\{V_t\}_{t\in \mathbb{R}}$?
- How similar are the flows if not the same? For example if I have theorems that hold for (1) do they hold for (2) and vice-versa?