Relative disk area is invariant under relative homotopy

For one normalized component $C$, write $U_{C,t}=U_C(\cdot ,t)$, ${\beta }_t=U_{C,t}^{\ast }\omega$, and define ${\alpha }_t\in {\mathrm{\Omega }}^1(C)$ by ${\alpha }_t(v)=\omega (d{U}_C({\mathrm{\partial }}_t),d{U}_C(v))$. The pullback $U_C^{\ast }\omega$ is closed by [F3]; writing it as ${\beta }_t+dt\wedge {\alpha }_t$ gives ${\mathrm{\partial }}_t{\beta }_t=d_C{\alpha }_t$ in local coordinates. To integrate $d_C{\alpha }_t$, use the following local compact-surface calculation. Cover compact oriented $C$ by finitely many oriented interior charts and, when $\mathrm{\partial }C\mathrm{\ne }\mathrm{\varnothing }$, boundary charts to the half-plane $y\ge 0$. Euclidean bump functions supported in these charts, normalized by their finite positive sum, give a finite smooth partition of unity. For each compactly supported chart form $\eta =Pdx+Qdy$, the interior formula ${\int }_{{\mathbb{R}}^2}d\eta =0$ follows by integrating ${\mathrm{\partial }}_{x}Q-{\mathrm{\partial }}_{y}P$ and using compact support; in a boundary chart, with outward-normal-first boundary orientation, ${\int }_{y\ge 0}d\eta ={\int }_{y=0}P(x,0)dx$ by the same one-variable fundamental theorem of calculus. Summing the localized identities proves ${\int }_{C}d\alpha ={\int }_{\mathrm{\partial }C}\alpha$ for disk components and ${\int }_{C}d\alpha =0$ for closed sphere components. Differentiation under the integral on compact $C$ now gives ${\int }_C{u}_{1,C}^{\ast }\omega -{\int }_C{u}_{0,C}^{\ast }\omega ={\int }_0^1{\int }_{\mathrm{\partial }C}{\alpha }_{t}dt$. This finite-chart proof uses no countable-choice assumption and applies to the full components, without deleting node markings.

On each boundary segment of a disk component, $U_C$ maps the swept segment into its fixed Lagrangian $L_i$. Both $d{U}_C({\mathrm{\partial }}_t)$ and $d{U}_C(v)$ for a boundary tangent $v$ therefore lie in $T{L}_i$, so ${\alpha }_t(v)=0$ by [F2]. Finitely many segment endpoints contribute no boundary integral, and closed sphere components have no boundary. The area change in step 1.1 is consequently zero on every normalized component; summing over the finite normalization from [F1] proves equality of total areas. Node matching keeps the homotopy a homotopy of the specified nodal maps, but no node-boundary cancellation is needed.