Lagrange multipliers and adiabatic limits II

In this second part to [FW22a] we finish the proof of the one-to-one correspondence of gradient flow lines of index difference one between the restricted functional and the Lagrange multiplier functional for deformation parameters of the metric close to the singular one. In particular, we prove that, although the metric becomes singular, we have uniform bounds for the Lagrange multiplier of finite energy solutions and all its derivatives. This uniform bound is the crucial ingredient for a compactness theorem for gradient flow lines of arbitrary deformation parameter. If the functionals are Morse we further prove uniform exponential decay. We finally show combined with the linear theory in part I that if the metric is Morse-Smale the adiabatic limit map is bijective. We present a general overview of the adiabatic limit technique in the article [FW22b].