Lagrangian Floer strip breaking and bubbling contract

Let ${\varphi }_t$ be the time-dependent evolution of $X_{H_t}$ with ${\varphi }_0=\mathrm{i}\mathrm{d}$, global by [F2]. Its symplectic preservation follows with the PHYSICS sign convention: $\frac{d}{dt}{\varphi }_t^{\ast }\omega ={\varphi }_t^{\ast }(d{\iota }_{X_{H_t}}\omega +{\iota }_{X_{H_t}}d\omega )={\varphi }_t^{\ast }(-d^2{H}_t)=0$. Thus no opposite-sign Hamiltonian definition is imported. Put $v_i(s,t)={\varphi }_t^{-1}{u}_i(s,t)$ and ${\tilde{J}}_t=(D{\varphi }_t{)}^{-1}{J}_{t}D{\varphi }_t$. Differentiation gives $(v_i{)}_s+{\tilde{J}}_t(v_i)(v_i{)}_t=0$ with boundaries $K_0=L_0$ and $K_1={\varphi }_1^{-1}{L}_1$. The structures remain compatible, and their boundary values at $t=0,1$ are fixed independently of $s,i$. Their compatible metric ${\tilde{g}}_t=\omega (\cdot ,{\tilde{J}}_t\cdot )$ is the pullback of $g_t$, so $E(u_i)=\int \mathrm{\mid }(v_i{)}_s{\mathrm{\mid }}_{{\tilde{g}}_t}^{2}dsdt=\int v_i^{\ast }\omega \mathrm{.}$ This equality is pointwise: $(v_i{)}_t={\tilde{J}}_t(v_i)(v_i{)}_s$. Fixed background Dirichlet energies are uniformly comparable with this energy on compact $[0,1]\times X$.

Apply [F3] to any translated sequence $v_i(s+a_i,t)$. Longitudinal translation preserves its exact equation and boundary coefficients. Uniform background-energy comparison supplies its required bound, and hence a finite global atom set and smooth local limit off it. The local puncture filling and actual disk/sphere rescalings in [F3] resolve every atom into a finite tree. Interior bubbles solve the autonomous ${\tilde{J}}_{\tau }$ equation at their original transverse coordinate $\tau$; physical boundary bubbles have boundary on $K_0$ or $K_1$. Every original Floer bubble is ${\varphi }_{\tau }$ composed with that bubble, since ${\varphi }_{t_i+{\rho }_{i}T}\to {\varphi }_{\tau }$ smoothly in the rescaling. It is consequently $J_{\tau }$-holomorphic and has the required original Lagrangian boundary when applicable. Symplectic preservation keeps its area unchanged and matches its value with the transformed base node.

For the finite-building branch, chords correspond bijectively to $p\in K_0\cap K_1$ via $x_p(t)={\varphi }_t(p)$. Flow conjugation identifies the specified path component with one homogeneous-boundary component. The intersection subset whose constant paths belong to that component is compact, and nondegeneracy makes it transverse and finite, by the component extension in [F4]. Degeneracy outside the relevant component is allowed. Apply [F4] to the homogeneous conjugated equation; all hypotheses of its relevant-component form now hold for arbitrary ${\tilde{J}}_t$. Fix a uniform $\delta >0$ below its small-window threshold. Pass to a subsequence whose total energies converge. If infinitely many maps are constant conjugated generators, fix that generator by a subsequence; all required conclusions are immediate. Otherwise every $v_i$ is nonconstant and $B_i=\{s:E(v_i;[s-2,s+2]\times [0,1])\ge \delta \}$ is a nonempty compact set: window energy is continuous, tends to zero at both ends by integrability, and emptiness would force constancy by [F4].

The energy identity here is the actual Floer/area identity, not only a comparable fixed-metric estimate. Around an atom at transverse coordinate $\tau$, replacing ${\tilde{g}}_t$ by ${\tilde{g}}_{\tau }$ changes the energy integral by at most $Cr$ times the bounded background energy in a radius-$r$ patch. This tends to zero as the patch shrinks. The root local tree identity in that frozen metric is its $\omega$-area identity for autonomous compatible bubbles; away from the atom, the base energy densities converge smoothly. Finite patch partition and shrinking therefore give precisely Floer base energy plus bubble areas on every bounded region with atom-free artificial edges. The same uniform comparison transfers zero neck energy/diameter. This proves the entire universal local branch for all original inputs, without any condition on other chords.

Starting from the least point of $B_i$, repeatedly select its least remaining point at least four units to the right. The selected length-four windows have disjoint interiors, each energy at least $\delta$, so there are at most $\lfloor E_0\mathrm{/}\delta \rfloor$ centers. Their rightward intervals of length four cover $B_i$ by the selection rule. Pass to a fixed count and then, for every successive center difference, a subsequence on which it stays bounded or tends to infinity. Group centers joined by bounded differences. Let $q_{\mathrm{\ell },i}$ be the first center of each group. Adjacent group translations diverge, and for some fixed $A>4$, every bad center lies in an inner core $(q_{\mathrm{\ell },i}-A+2,q_{\mathrm{\ell },i}+A-2)$. There are uniformly finitely many groups.

At each group translation apply the fully proved local construction of steps 2.1–3.1 on a countable exhaustion of the strip. This gives a finite-energy strip $v_{\mathrm{\ell }}$ with globally finitely many sphere/disk trees and actual rescalings, matching and stability. Finitely many subsequences retain all levels together. Each level has positive total energy at least $\delta$: its selected central window has at least that energy, and a larger fixed core with atom-free vertical edges captures it under the exact local energy identity. This does not assume energy convergence across an atom or discard a constant base carrying bubbles.

In every gap $[q_{\mathrm{\ell },i}+A,q_{\mathrm{\ell }+1,i}-A]$ and on both outer half-infinite tails, all length-four window centers have energy below $\delta$; all source buffers exist because the original strips are defined on the whole strip. Apply [F4]. For every fixed $R>A+1$ and all large $i$, the energy outside disjoint cores $[q_{\mathrm{\ell },i}-R,q_{\mathrm{\ell },i}+R]$ is at most $CM\delta e^{-c(R-A)}$, where $M$ is the number of levels. On each central gap and outer tail, image diameter and distance from its one generator have the corresponding $C\sqrt{\delta }{e}^{-c(R-A)\mathrm{/}2}$ bound. Fix those finitely many generator choices by a subsequence. The positive end of $v_{\mathrm{\ell }}$ and the negative end of $v_{\mathrm{\ell }+1}$ are this same generator, by passing those estimates to fixed translated coordinates and then sending them to infinity. The outer estimates give the original endpoint generators, since the original ends are $x_{\pm }$ after undoing conjugation. Thus every end matches; no concentration is silently discarded at an end.

Choose $R\to \mathrm{\infty }$ through radii whose vertical edges avoid every level's finite atom set. For each fixed $R$, add the finitely many exact local core energy identities from step 3.1. The omitted energy is bounded by step 5.1 and tends to zero as $R\to \mathrm{\infty }$. Each base energy exhausts its strip and every finite atom eventually lies in a core. Hence ${\lim }_{i}E(u_i)={\sum }_{\mathrm{\ell }}E(u_{\mathrm{\ell }})+{\sum }_v\int u_v^{\ast }\omega \mathrm{.}$ The same estimates give zero neck/tail diameter in conjugated coordinates. Returning by ${\varphi }_t$ gives convergence to the matching Hamiltonian chord, rather than falsely asserting that an entire chord has zero diameter in $X$. Uniform positive bubble quanta from [F3] bound the nonconstant bubble count by total energy divided by that quantum; its finite-tree counting bounds the ghost count by three times that number. The level count was already bounded in step 3.2.

Finally put $w={\mathrm{\partial }}_{t}u-X_{H_t}(u)$ on any original strip level. Its equation gives ${\mathrm{\partial }}_{s}u=-J_{t}w$, hence $\mathrm{\mid }{\mathrm{\partial }}_{s}u{\mathrm{\mid }}_{g_t}^2=\omega ({\mathrm{\partial }}_{s}u,w)$. By [F1], $\frac{d}{ds}{\mathcal{A}}_H([u(s,\cdot ),{\mathrm{c}\mathrm{a}\mathrm{p}}_s])=-\int \mathrm{\mid }{\mathrm{\partial }}_{s}u{\mathrm{\mid }}_{g_t}^{2}dt$. Integrate to its ends, which are exponentially convergent by [F4] on the finite branch (and by the same local chart argument at the two specified nondegenerate original ends). This gives exactly its action drop. Caps transported along individual levels give their lifts; adjacent lifts may differ by bubble classes, whose symplectic areas are precisely the bubble terms already retained in step 6.1. No reduction of coefficient rings, orientation/gluing, virtual Stokes or A-infinity conclusion is involved.