Witten-Helffer-Sjostrand Theory for a Generalized Morse Functions

Abstract

In this paper, we extend the Witten-Helffer-Sj\"{o}strand theory from Morse functions to generalized Morse functions. In this case, the spectrum of the Witten deformed Laplacian $\Delta(t)$, for large t, can be seperated into the small eigenvalues (which tend to 0 as $t\rightarrow\infty$), large and very large eigenvalues (both of which tend to $\infty$ as $t\rightarrow\infty$). The subcomplex $\Omega_{0}^{\ast}(M,t)$ spanned by eigenforms corresponding to the small and large eigenvalues of $\Delta(t)$ is finite dimensional. Under some mild conditions, it is shown that $(\Omega_{0}^{\ast}(M,t),d(t))$ converges to a geometric complex associated to the generalized Morse function as $t\rightarrow\infty$.