the theorem that follows and throughout the note we take eigenvalues of the metric $g_Y$ with respect to the euclidean metric ${\overline{g}}_i$ in normal coordinates for the $V_i$. We let $\Omega_i$ denote the maximum eigenvalue of $g_Y(y,0)$ in each $\overline{W_i}$, and then set $\Omega = \max_i \; \Omega_i$. Similarly, let $\omega$ be the minimum eigenvalue of $g_Y(y,0)$ over the cover $\overline{W_i}$. As this covering and constants will be used throughout the paper, we always use normal coordinates along $Y$ in any choice of Fermi coordinates that follows.

An adaptation of the comparison theorems in [@Petersen] yields the following theorem.

Let $(y^{\beta}, r)$ be Fermi coordinates for $Y$ on $W_i \times [0,\infty)$ for an open set $W_i$ in the reference covering described above. Let $\Lambda, \lambda$ denote the maximal and minimal eigenvalues of the shape operator over $Y$. We have: \[thm:comparison\]

**Shape operator estimate:** $$\begin{aligned}
\label{eqn:shapeinequality}
 a \tanh( a( r + L_1) ) \; \delta^{\beta}_{\gamma} \; \leq \; S^{\beta}_{\gamma}(y, r) \; \leq \; b \coth( b( r + L_2) ) \; \delta^{\beta}_{\gamma},\end{aligned}$$ where $L_1 = \frac{1}{a} \tanh^{-1}(\frac{\lambda'}{a})$, $L_2 = \frac{1}{b} \coth^{-1}(\frac{\Lambda'}{b})$, and $$\begin{aligned}
\label{eqn:constants}
\Lambda' = \begin{cases}  \Lambda  & \text{ if } \Lambda > b, \\
         2b & \text{ if } 