Small local energy forces cylinder neck collapse

Fix the patch radius $r_0$ in the statement, decreasing it if necessary to satisfy the uniform scale restrictions of [F1]. Every point in $[a,b]\times D$ is covered by an interior smaller patch or a boundary half-radius patch of this radius, contained in the extended cylinder. In the angular interval case, points within $r_0\mathrm{/}4$ of an edge use the boundary supremum estimate centered at their boundary foot; other points use an interior disk of radius at least $r_0\mathrm{/}8$. Their energies are at most $\delta$: if their radii are smaller, each is contained in one stipulated radius-$r_0$ patch. In the angular circle case all points use an interior disk. Choosing ${\delta }_0$ below the [F1] thresholds gives ${\sup }_{[a,b]\times D}\mathrm{\mid }du{\mathrm{\mid }}_g\le C_0\sqrt{\delta }\mathrm{.}$ The unit buffer ensures these patches remain in the prescribed source domain even at $\sigma =a,b$.

Choose ${\delta }_0$ further so that $C_0\sqrt{{\delta }_0}$ is below half the small-gradient threshold in [F2], and choose ${\kappa }_0$ below half its source-variation threshold. All hypotheses of the corresponding neck estimate hold on $[a,b]\times D$. In the boundary case the small derivative bound also places every angular half-loop in the fixed reflection collar, since it starts on $L$ and has length at most $T{C}_0\sqrt{\delta }$. Thus the original periodic embedding, or the reflected periodic double in [F2], is genuinely defined. Its angular period is $T$ or $2T$, and its endpoint slice energies obey $e(a)+e(b)\le C_1\delta$, by the derivative estimate in step 1.1 and the fixed differential bounds of $F$ and $\rho$.

The exponential barrier in [F2] therefore gives $e(\sigma )\le C_1\delta (e^{-c(\sigma -a)}+e^{-c(b-\sigma )})\mathrm{.}$ Its integral on the central interval is at most $C\delta e^{-cR}$, proving the asserted energy bound. The same inequality and $\sqrt{x+y}\le \sqrt{x}+\sqrt{y}$ give ${\int }_{a+R}^{b-R}\sqrt{e(\sigma )}d\sigma \le C\sqrt{\delta }{e}^{-cR\mathrm{/}2},{\sup }_{[a+R,b-R]}\sqrt{e}\le C\sqrt{\delta }{e}^{-cR\mathrm{/}2}\mathrm{.}$ These bounds do not multiply by cylinder length, since the exponential tails are integrated explicitly.

Let $P$ be the fixed angular period of $V$, and $B(\sigma )=P^{-1}\int V(\sigma ,\tau )d\tau$. Angular integration and Holder give $\mathrm{\mid }V(\sigma ,\tau )-B(\sigma )\mathrm{\mid }\le C\sqrt{e(\sigma )}\mathrm{.}$ Moreover $\mathrm{\mid }{V}_{\sigma }\mathrm{\mid }\le C\mathrm{\mid }{V}_{\tau }\mathrm{\mid }$ on each side, by the original equation, bounded $J$, and the fixed reflection differential and its inverse. Thus $\mathrm{\mid }{B}^{\mathrm{\prime }}(\sigma )\mathrm{\mid }\le P^{-1}\int \mathrm{\mid }{V}_{\sigma }\mathrm{\mid }d\tau \le C\sqrt{e(\sigma )}\mathrm{.}$ For any two central source points, compare their embedded values to their slice averages and integrate this bound for $B^{\mathrm{\prime }}$ between the slices. Step 3.1 proves that the Euclidean diameter of their embedded images is at most $C\sqrt{\delta }{e}^{-cR\mathrm{/}2}$. This uses no cancellation of angular endpoint terms; the reflected periodic map is used only after its actual construction in [F2].

Finally the inverse of the fixed compact embedding is uniformly locally Lipschitz in ambient chord distance. Indeed in each smaller image chart its inverse derivative is bounded; compactness gives a uniform closeness scale within one chart. For pairs separated by more than that scale, the diameter of each relevant compact target component enlarges the constant to a global bound. The connected image of our cylinder lies in one target component, and a compact manifold has finitely many components. Consequently $d_g(x,y)\le C\mathrm{\mid }F(x)-F(y)\mathrm{\mid }$ for pairs in that component, with uniform constant. Applying this to step 4.1 gives the asserted target diameter estimate. Hence both energy and image collapse with $R$ as claimed, conditional only on the concrete local-energy and source-variation bounds in the statement.