Uniform stationary return persists under arbitrary tangential drift

If $s_0=0$, take $t_{\ast }=0$. Otherwise set $\sigma =-\mathrm{sgn}(s_0)$ and follow $p(u)={\mathrm{\Phi }}_{\sigma u}^T(s_0,{\rho }_0,y_0)$ for $u\ge 0$ until the first meeting with $s=0$. Set $T_{ϵ}=4ϵ\mathrm{/}c$. Choose $ϵ$ so that $B{T}_{ϵ}<\log 2$, $2ϵ<\eta$, and $T_{ϵ}$ is strictly within a common local-flow interval for the compact reference collar. Such an interval follows by taking a finite subcover of the open flow domain around that collar. While the trajectory remains in the wedge before meeting, $\mathrm{\mid }d\rho \mathrm{/}du\mathrm{\mid }\le B\rho$ and [F3] applied to $\log \rho$ give $e^{-Bu}{\rho }_0\le \rho (u)\le e^{Bu}{\rho }_0$. Since $ds\mathrm{/}du=\sigma a$, the sign of $s$ before meeting gives $d\mathrm{\mid }s\mathrm{\mid }\mathrm{/}du=-a<0$. Thus $\mathrm{\mid }s\mathrm{\mid }\mathrm{/}\rho \le 2ϵ<\eta$. Taking the fixed starting collar with ${\rho }_0<r\mathrm{/}4$ gives $0<\rho (u)<r\mathrm{/}2$ for $u\le T_{ϵ}$. The strict wedge and radial bounds, and the common flow interval, exclude exit before either meeting or time $T_{ϵ}$. This also excludes reaching $\rho =0$ in that time.

If there were no meeting for $0\le u\le T_{ϵ}$, step 1.1 would give $a\ge (c\mathrm{/}2)\rho \ge c{\rho }_0\mathrm{/}4$ throughout. Integrating $d\mathrm{\mid }s\mathrm{\mid }\mathrm{/}du=-a$ by [F3] would yield $\mathrm{\mid }s(T_{ϵ})\mathrm{\mid }\le \mathrm{\mid }{s}_0\mathrm{\mid }-(c{\rho }_0\mathrm{/}4)T_{ϵ}\le 0$, contradicting absence of a meeting. Therefore the first meeting has $t_{\ast }=\sigma u_{\ast }$ with $0\le u_{\ast }\le T_{ϵ}$, and the radial comparison follows from step 1.1. Since the inequalities have strict margins for sufficiently small $ϵ$, this return construction applies in an open neighborhood of each point under consideration with $\rho >0$.

In local initial-point coordinates, put $H(t,p)=s({\mathrm{\Phi }}_t^T(p))$. At the meeting, ${\mathrm{\partial }}_{t}H=a({\mathrm{\Phi }}_{t_{\ast }}^T(p))\ge (c\mathrm{/}2)\rho (t_{\ast })>0$. Smooth flow dependence and the implicit function theorem therefore give a smooth local root $t=t_{\ast }(p)$, and composition gives the smooth return point. These local roots agree with the constructed return: $a>0$ makes the $s$ coordinate strictly increasing along the connected trajectory segment inside the wedge, so that segment has at most one zero. The same argument applies at initial points with $s_0=0$, so there is no smoothness exception there. Along the entire segment, the moving sphere coordinate has speed at most $M$ and hence path length at most $M\mathrm{\mid }{t}_{\ast }\mathrm{\mid }\le M{T}_{ϵ}$ in the fixed sphere metric. No tangential constancy is used; all normal bounds hold at the moving base point. The bounds and choices of collar and time depend only on the fixed reference coefficients, so they are uniform for graphs satisfying the same pointwise smallness bound.

Under the additional exterior hypotheses, choose the smaller coefficient wedge furnished by [F1] inside $\mathcal{E}$ before making the choices above. Steps 1.1 and 2.1 keep the entire return segment in that wedge, so it stays in the exterior collar. The return and radial estimates and step 3.1 require only the coefficient collar and remain valid without the additional exterior hypotheses.