Morse theory for Lagrange multipliers and adiabatic limits

TL;DR

This paper explores Morse homology for a Lagrange multiplier function on a manifold, analyzing how it behaves under large and small rescaling of the metric, linking it to the topology of the constraint set.

Contribution

It establishes a connection between Morse homology for the Lagrange multiplier function and the topology of the constraint set via adiabatic limits, using geometric and analytical techniques.

Findings

For large mbda, the Morse complex matches that of the original function with a shifted grading.

As mbda approaches zero, the Morse complex relates to the homology of the constraint set.

Homotopy invariance of the Morse homology under mbda variation is demonstrated.

Abstract

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 $M \times {\mathbb R}$, and rescale its ${\mathbb R}$-component by a factor $\lambda^2$. We show that generically, for large $\lambda$, 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.