Exponential bound on partition functions for long-cylinder degenerations

Prove that for families of closed surfaces obtained by gluing spheres with three holes via long thin cylinders, the partition function Z(Σ) times exp(−(c/6)·l_i·λ_min) is uniformly bounded independently of the surface Σ, where λ_min is the minimal positive eigenvalue of L_0+L̄_0 and l_i are cylinder lengths.

Background

In the collapse analysis via Segal’s axioms, surfaces are decomposed into pairs of pants joined by long cylinders; spectral data of L_0+L̄_0 govern decay along tubes.

The conjectured bound would control partition functions uniformly under the double-scaling limit, providing quantitative input for compactness and convergence arguments.