I'm trying to understand why in the following lemma (Lemma 10.7 of [BEHWZ]), the upper bound on the $L^{\infty}$-norm of the differential is given in terms of the injective radius w.r.t to a specific metric on the domain.
Let me give the full statement of the Lemma here and explain the notation:
Lemma 10.7 There exists an integer $K=K(E)$ which depends only on the energy bound $E$ such that, by adding to each marked point set $M_n$ a disjoint set $Y_n$ of cardinality $2K$, we can arrange a uniform gradient bound $$ \| \nabla F_n(x) \| \leq \dfrac{C}{\rho(x)} \ x \in \dot{S_n} \setminus Y_n $$ where the gradients are computed with respect to the cylindrical metric on $\Bbb R \times V$ associated with a fixed Riemannian metric $V$, and the hyperbolic metric on $\dot{S_n}\setminus Y_n$, and where $\rho(x)$ is the injectivity radius of this hyperbolic metric at the point $x \in \dot{S_n}\setminus Y_n$
$S_n$ is a closed Riemann surface, $F_n$ is a pseudoholomorphic map defined on $S_n \setminus Z_n$, where $Z_n$ is some discrete set of points that act as puncture. We assume that, up to add some more punctures, we have an hyperbolic metric on $\dot{S_n}:= S_n\setminus Z_n$.
Why can't they prove, with the very same proof, the existence of some uniform bound independent of the injectivity radius? Since this is a stronger statement it could be that it doesn't apply in this case but it's not clear to me where exactly in the proof we have to introduce the injectivity radius.
I feel it might be because the metric they're working with might "change" when doing the usual rescaling argument to have a sphere/plane bubbling off. I used the word "change" because I'm not able to make it more precise and I'd like to understand it better.
Thanks in advance for any insights on this proof.
REFERENCES
[BEHWZ] Bourgeois, F.; Eliashberg, Y.; Hofer, H.; Wysocki, K.; Zehnder, E. Compactness results in symplectic field theory. Geom. Topol. 7 (2003), 799--888.
