along $\gamma_p$. Explicitly, $$J(t) = \dot\sigma^{\beta}(0) \partial_{\beta} |_{(\sigma(0),t)}.$$ So these special Jacobi fields have constant components in Fermi coordinates. Convexity of $Y$ easily implies the following estimates:

Let $J(t) = \dot\sigma^{\beta}(0) \partial_{\beta} |_{(\sigma(0),t)}$ be the Jacobi field along $\gamma$ described above. Then

1.  ${\left\langle  J(0) \;, \; D_t J(0) \right\rangle} \geq 0$,

2.  $|J(t)| \geq |J(0)| \; \cosh(at)$.

\[lemma:jacobigrowth\]

The comparison theorems we use are based on the treatment given in [@Petersen]. These are obtained by analysis of the Riccati differential equation . In what follows an inequality involving the shape operator of the form $S \geq c$ means that every eigenvalue of $S$ is greater than or equal to $c$. Inequalities involving a metric are to be interpreted as inequalities between quadratic forms.

For the metric comparisons that follow we require a covering of the compact hypersurface $Y$. Fix $\epsilon = \frac{1}{2} \min\{ \operatorname{inj}(Y), \pi \}$, where $\operatorname{inj}(Y)$ is the injectivity radius of $g_Y(y,0)$. For any $y \in Y$, the ball $B^Y_{\epsilon}(y)$ is the domain of a convex normal coordinate chart with image $B_{\epsilon}(0) \subset {\mathbb{R}}^n$. On the ball $B_{\epsilon}(0)$, we will need to consider three metrics, the original $g_Y$ (transfered to $B_{\epsilon}(0)$ by means of normal coordinates), the 