Morse theory for Lagrange multipliers and adiabatic limits

Given two Morse functions $f,\mu$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M\times \mathbb{R}$ which sends $(x,\eta )$ to $f(x)+\eta \mu (x)$. Take a product metric on $M \times {\mathbb R}$,andrescaleits${\mathbb R}$-componentbyafactor$\lambda^2$\mathrm{.}Weshowthatgenerically,forlarge$\lambda$, the Morse-Smale-Witten chain complex is isomorphic to the one for $f$ and the metric restricted to ${\mu }^{-1}(0)$, with grading shifted by one. On the other hand, let $\lambda \to 0$, we obtain another chain complex, which is geometrically quite different but has the same homology as the singular homology of ${\mu }^{-1}(0)$ and the isomorphism between them is provided by the homotopy by varying $\lambda$. Our proofs contain both the implicit function theorem on Banach manifolds and geometric singular perturbation theory.