Let $X:\mathbb{R}^n \to \mathbb{R}^n$ be a smooth and bounded vector field, such that $$ X_n \ge c|(X_1, \dots, X_{n-1})| \ge \epsilon > 0\,. $$ Under these assumptions one can prove that integral curves $\Phi(\cdot, x)$ exist for all times (and are unique) and they intersect every hyperplane of fixed $n$-th coordinate exactly once.
Thus, fixed the hyperplane $H_{h} := \{x\,:\,x_n = h\}$, we have that the flow $\Phi : \mathbb{R} \times H_h \to \mathbb{R}^n$ is a bijection (I "foliate" $\mathbb{R}^n$ with Lipschitz curves).
Fix a bounded smooth set $A \subset H_h$ (wlog a ball of $\mathbb{R}^{n-1}$). I want to look at this set as it gets transformed by the flow $\Phi$ generated by $X$. More precisely, fix another hyperplane $H_k$, with $k<h$, and define the set $$ C:= \{x\,:\, k\le x_n \le h\}\cap \bigcup_{x\in A} \Phi([-T,T],x) \,, $$ where $T:= \inf_t\{\Phi([-t,t],x) \cap H_k \neq \emptyset\,, \forall x\in A \}$. Under our working hypotheses this guy is well defined since all integral curves reach $H_k$ in finite time.
Is it true that the "lateral surface" of this set is regular (define $\partial C$ in the same way as $C$ by taking the union over $x\in \partial A$ rather than in $A$)? Can I say it is diffeomorphic to a cylinder?
How do I go about proving it?