Relative path-shortening connectivity from index bounds

For a continuous compact family $K\to C^0([0,1],P)$, joint continuity on $K\times [0,1]$ gives a uniform mesh whose edges lie in convex balls. Replace each edge by its unique shortest geodesic and interpolate pointwise using the short geodesic joining the old and new values. With a sufficiently fine mesh this fixes all constant-speed minimizing geodesics, so fixes the prescribed map on $A$. The resulting polygon energy has a finite bound $c$ by compactness. Refine the mesh by a multiple until $n>c\mathrm{/}{r}^2$, where $r$ is a convexity radius. This refinement changes neither the polygon paths nor their energy, and places the family in the compact polygon sublevel of [F1].

The same operations preserve $H^1$ and depend continuously in its topology. On a convex coordinate ball they are compositions with smooth maps whose first two derivatives are bounded on a compact neighborhood. The one-dimensional estimate $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^0}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}$ and the chain rule give continuity: the derivative difference is bounded by a constant times the $H^1$ difference plus the $C^0$ difference times a bounded $L^2$ derivative. The estimate follows by writing $u(t)=u(s)+{\int }_s^t{u}^{\mathrm{\prime }}$ and choosing $s$ with $\mathrm{\mid }u(s)\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}$. Compact $H^1$ families therefore have a uniform convex mesh. On an interval of length $1\mathrm{/}n$, Cauchy–Schwarz gives $n\mathrm{dist}(x_{j-1},x_j{)}^2\le \int \mathrm{\mid }{\gamma }^{\mathrm{\prime }}{\mathrm{\mid }}^2$, so their polygon energies stay bounded.

Increase $c$ slightly to a regular value and refine the mesh if needed so that [F1] still gives a compact sublevel. List the finitely many critical values below this bound and choose regular levels separating them, enclosing every component at the same value in the same crossing band. The polygon domain is a finite-dimensional boundaryless manifold and all these closed bands are compact by [F1], so the precise crossing hypotheses of [F2] hold. Apply [F2] successively from the highest nonminimum level downwards. Every nonminimum handle has negative rank at least $d+1$, so the family can bypass it relative to $A$. It reaches a small minimum neighborhood, whose normal contraction takes it to the minimum locus and fixes $A$. The endpoint path component and its minimum energy are retained throughout. This proves the claimed finite-CW relative connectivity in both topologies, without any assertion of a global deformation retraction or continuously convergent infinite-time flow.