Pollicott–Ruelle spectrum and Witten Laplacians

  • Nguyen Viet Dang

    Université Claude Bernard Lyon 1, Villeurbanne, France
  • Gabriel Rivière

    Université de Lille and Université de Nantes, France
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We study the asymptotic behavior 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 spectral data of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. 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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Nguyen Viet Dang, Gabriel Rivière, Pollicott–Ruelle spectrum and Witten Laplacians. J. Eur. Math. Soc. 23 (2021), no. 6, pp. 1797–1857

DOI 10.4171/JEMS/1044