Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $(g\_t)\_{0\leq t\leq T}$ be a family of Riemannian metrics with $g\_0=g$. Suppose that $\partial M$ has at least two connected components. Suppose that there exists $C>0$ such that $|Rm(g\_t)|\_{g\_t}\leq C$ and $|\mathcal{A}(g\_t)|\_{g^T\_t}\leq C$ for all $0\leq t\leq T$. Here $\mathcal{A}(g\_t)$ is the second fundamental form of $\partial M$ and $g^T\_t$ the restriction of $g\_t$ to $\partial M$. Is there $c>0$ such that the distance of two components of $(\partial M,g^T\_t)$ is bounded below by $c$ (uniformly in $t$)?



Control distance of the boundary
No, consider a part of Lobachevsky plane bounded by two asymptotically parallel lines.


