A Thom-Smale-Witten theorem on manifolds with boundary

TL;DR

This paper proves a canonical isomorphism between the small eigenvalue eigenspaces of the Witten Laplacian and the Thom-Smale complex on manifolds with boundary, extending Morse theory via analytic localization techniques.

Contribution

It establishes a Thom-Smale-Witten theorem for manifolds with boundary, linking spectral analysis to Morse theory in this setting.

Findings

Isomorphism between small eigenvalue eigenspaces and Thom-Smale complex

Extension of Morse inequalities to manifolds with boundary

Application of Bismut-Lebeau's localization techniques

Abstract

Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization techniques. As a by-product, we obtain Morse inequalities for manifolds with boundary.