Lemma: Uniform metric and localization estimates

For every $\mu \in I$,

Define

Then

The region $s+k\ge h$, for $0<s\le 1$, satisfies

The graph is horizontal, $\tau =s+2$, whenever $\rho \ge r$.

Proof. Write $\vartheta /2=\pi /4+t$, so $\left|t\right|\le \pi /120<27/1000$, and

The bounds $\cos t\ge 1-t^2/2$, $\cos t\le 1$, and $\left|\sin t\right|\le \left|t\right|$ give (eq:ab-bounds). The bound on $b$ follows from $0<\vartheta <\pi$. Thus $d>a-3/10>69/200$. Because $b<a$, $d>a/2$, which gives $1/d<2/a=r$ and $\chi <\beta$. Also $r<400/129<7/2$.

Since $\min \{\xi ,\eta \}\le \rho /2$, $h-k\ge d\rho$. Dominance of $s+k$ therefore implies $\rho \le s/d$. For $\rho \ge r$, $h\ge 2$, so the cap in (eq:family) is active.