Morse-Bott functions and the Witten Laplacian

[摘要] Given a compact Riemannian manifold (N, g), a flat vector bundle V over N, and a Morse-Bott function h, Witten considered the following one-parameter deformation of the differential d in the de Rham complex of V-valued differential forms on N:$$d(alpha ):omegamapsto esp{-alpha h}desp{alpha h}.$$In this thesis we study the asymptotics as $alphaoinfty$ of the discrete spectrum of the Witten Laplacian$$L(alpha)=d(alpha )dsp{*}(alpha )+dsp{*}(alpha )d(alpha ).$$ Suppose g is a metric on N, associated to a Morse-Bott function h. The main result of the thesis states that as $alphaoinfty$ the small eigenvalues of $L(alpha)$ approach all eigenvalues of the standard Laplacian $Delta$ on M, twisted by the orientation bundle of the negative directions in the normal bundle to M in N. We also prove the estimates on the rate of convergence as $alphaoinfty$ of the small eigenvalues of $L(alpha ).$ The main idea of the proof is to use the adiabatic limit technique of Mazzeo-Melrose and Forman to analyze the spectrum of the Witten Laplacian on the tubular neighborhood of the critical submanifold M of h. As an application of our results we give a new Hodge theoretic proof of the Thom isomorphism. We obtain the localization of the trace of the heat kernel $esp{-tL(alpha})$ along the critical submanifold M of the Morse-Bott function h, and we prove the degenerate inequalities of Morse.

[发布机构] Rice University