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Article Dans Une Revue Journal of the European Mathematical Society Année : 2019

Pollicott-Ruelle spectrum and Witten Laplacians

Gabriel Riviere

Résumé

We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey–Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten-Helffer-Sjöstrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.
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Dates et versions

hal-01585774 , version 1 (12-09-2017)
hal-01585774 , version 2 (18-06-2019)

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Nguyen Viet Dang, Gabriel Riviere. Pollicott-Ruelle spectrum and Witten Laplacians. Journal of the European Mathematical Society, In press, ⟨10.4171/JEMS/1044⟩. ⟨hal-01585774v2⟩
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