\section{Chute de curvature equations}

\begin{definition} Under the conditions of Lemma
\ref{defnormalflow}, for every $t\in J$ we can parameterize the level line of $u(t,\fxx)$ passing by $\fxx(t,\fxx_0)$
by its length $s$, in such a way that the parameterization $s\to \tilde \fxx(t,s)$ satisfies
$\tilde\fxx(t,0)=\fxx(t,\fxx_0)$ and $\frac{\partial}{\partial s}\tilde\fxx(t,0)=\frac{Du^\perp}{|Du|}(t,
\fxx(t,\fxx_0)).$ In that way, we get a local parameterization $\tilde\fxx(t,s)$ of the level lines of $u$ passing in a
neighborhood of $(t_0, \fxx_0)$. This parameterization is uniquely associated with $u$ and $(t_0,\fxx_0)$. We call
$\tilde\fxx(t,s)$  the {\rm normal parameterization of the level lines of $u$} around $(t_0,\fxx_0)$.
\end{definition}

\begin{proposition}\label{corollary5.9}
Assume that the function $(t,\fxx)\mapsto u(t,\fxx)$ is $C^2$ in a neighborhood of $(t_0,\fxx_0)$ and that
$Du(t_0,\fxx_0)\not=0$. Then $u$ satisfies the curvature motion equation
\begin{equation}\label{eq68x}
\frac{\partial u}{\partial t}(t,\fxx)= {\rm curv}(u)(t,\fxx)|Du|(t,\fxx)
\end{equation}
in a neighborhood of $(t_0,\fxx_0)$ if and only if the normal parameterization of the level lines of $u$ passing in
this neighborhood, $(t,s)\mapsto \tilde\fxx(t,s)$ satisfies the intrinsic heat equation \eqref{eq61x},
\[
\frac{\partial\tilde\fxx}{\partial t}(t,s)=\frac{\partial^2\tilde\fxx}{\partial s^2}(t,s)=\bm{\kappa}(\tilde\fxx)(t,s).
\]
\end{proposition}

 \begin{proof} We shall use Relation \eqref{eq62x}, which
 establishes the link between $curv(u)(\fxx)$ and  the curvature
 $\bm{\kappa}(\fxx)$ of the level line passing by $\fxx$.
If $\tilde \fxx$ satisfies
 \begin {equation}\label{eq61xtilde}
\frac{\partial\tilde\fxx}{\partial t}(t,s)=\frac{\partial^2\tilde\fxx}{\partial s^2}(t,s)=\bm{\kappa}(\tilde\fxx)(t,s),
\end{equation}
by (\ref{eq66x}), we get
$$-(\frac{\partial u}{\partial t}\frac{Du}{|Du|^2})(t,\fxx(t))=
 \bm{\kappa}(\fxx(t)).$$
By (\ref{eq62x}), this yields
$$(\frac{\partial u}{\partial t}\frac{Du}{|Du|^2})(t,\fxx(t))=
{\rm curv(u)}\frac{Du}{|Du|},$$ which implies the curvature motion equation (\ref{eq68x}). Conversely, substituting in
(\ref{eq66x}) the value of $\frac{\partial u}{\partial t}$ given by (\ref{eq68x}) yields
$$\frac{\partial \fxx}{\partial t} = -({\rm curv}(u)|Du|\frac {Du}{|Du|^2})$$
and using (\ref{eq62x}) we obtain the heat intrinsic equation (\ref{eq61x}), $\frac {\partial \fxx}{\partial t}(t,s)=
\bm{\kappa}(\fxx(t))$.
\end{proof}
