Let $M$ be a smooth manifold of $n$ dimensional with no boundary and $\varphi:M\times[0,T)\to\mathbb{R}^{n+1}$ be a smooth solution of the mean curvature flow \begin{align} \dfrac{\partial\varphi_t}{\partial t}=\Delta_t \varphi_t\ \ \ \text{ on }M\times[0,T). \end{align} or a smooth family of immersions from $M$ to $\mathbb{R}^{n+1}$ moving by mean curvature.
Then are either of them true that:
(i) the immersion $\varphi_t:M\to\mathbb{R}^{n+1}$ remains to be proper if the initial data $\varphi_0$ is proper.
(ii) the metric $g_t$ on $M$ induced from the Euclidean metric of $\mathbb{R}^{n+1}$ by $\varphi_t$ remains to be complete if $g_0$ is complete.
? Here a map $f:X\to Y$ between topological spaces $X, Y$ is said to be proper iff \begin{align} K\stackrel{\mathrm{compact}}\subset Y\Rightarrow f^{-1}(K)\stackrel{\text{compact}}\subset X. \end{align}
Hence if an immersion $\iota:M\to\mathbb{R}^{n+1}$ is proper, then it induces a complete metric on $M$. Thank you.