I'm studying the shortening process, introduced in the book Course in Minimal Surfaces by T. Colding and W. Minicozzi, which is inspired by the Birkhoff's curve shortening process.
In the book, is defined the curve shorting map $\Psi$ by certain steps of the process which can be found on pages 165-166. The authors define the function in the space $\Lambda$, which is defined as the set of piecewise geodesics with exactly $L$ breakpoints, such that the length of each geodesic segment is at most $2\pi$ and parameterized proportionally by the arc length, with Lipschitz constant $L$. For more details see here.
One of the properties of this map is that it is continuous, which is my main question. Given that the map $\Psi: \Lambda \to \Lambda$ is defined in $\Lambda$, but to proof the continuity, the following lemma is used
Let $\gamma:S^1\to M$ be a $W^{1 ,2}$ map with $E(\gamma) <L$. If $\gamma_e$ and $\tilde{\gamma}_e$ are given by applying steps $(A_1)$ and $(B_1)$ to $\gamma$ then the map $\gamma\to \tilde{\gamma}_e$ is continuous from $W^{1,2}$ to $\Lambda$ equipped with the $W^{1 ,2}$ norm.
In the lemma, we take a curve $\gamma$ on $W^{1,2}$ (I know that $\Lambda$ is a subset of $W^{1,2}$). But this is my hesitation, why in the lemma it takes a curve in $W^{1,2}$, if $\Psi$ is defined in $\Lambda$?
I know continuity in $W^{1,2}$ will imply continuity in $\Lambda$, but then why hasn't the map already been defined in $W^{1,2}$?
In fact, I imagined, that the map should have already been defined in $W^{1,2}$, to have a greater amount of curves that we can apply the process, but when he says that the map $\Psi:\Lambda\to\Lambda$ is continuous, and then when he proofs this property he uses the $W^{1,2}$ space, I end up getting lost...
Could someone help me understand (and who knows, how to interpret) this $\Psi$ map?
I have provided screamshots of the relevant pages in the book, they are linked in the text.