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×R, which sends (x,η) to f(x)+ημ(x). Take a product metric on M×R, and rescale its R-component by a factor λ2. 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, in the limit λ→0, we obtain another chain complex, which is geometrically quite different but has the same homology as the singular homology of μ−1(0). The isomorphism between the chain complexes is provided by the homotopy obtained by varying λ. Our proofs use both the implicit function theorem on Banach manifolds and geometric singular perturbation theory.