Morse theory for Lagrange multipliers and adiabatic limits

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Schecter, Stephen¹  Xu, Guangbo^2,3 

[1] N Carolina State Univ, Dept Math, Raleigh, NC 27695 USA

[2] Princeton Univ, Dept Math, Princeton, NJ 08544 USA

[3] Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA

关键词: Morse homology;    Geometric singular perturbation theory;    Exchange lemma;    Adiabatic limit;   

DOI  :  10.1016/j.jde.2014.08.018

来源: Elsevier

【 摘 要 】

Given two Morse functions f, mu, on a compact manifold M, we study the Morse homology for the Lagrange multiplier function on M x R, which sends (x, eta) to f (x) + eta mu(x). Take a product metric on M x R, and rescale its 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, in the limit lambda -> 0, we obtain another chain complex, which is geometrically quite different but has the same homology as the singular homology of mu(-1)(0). The isomorphism between the chain complexes is provided by the homotopy obtained by varying lambda. Our proofs use both the implicit function theorem on Banach manifolds and geometric singular perturbation theory. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free