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Morse theory for Lagrange multipliers and adiabatic limits

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