Small eigenvalues of the Witten Laplacian acting on p-forms on a surface

Abstract

In this article, we are interested in the exponentially small eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian associated with some Morse function, in the general framework of p-forms, on a connected compact Riemannian manifold without boundary. Our purpose is to notice that the knowledge of (the asymptotic formulae for) the smallest non-zero eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian acting on functions, presented by Helffer, Klein and Nier in Matematica Contemporanea 26 (2004), 41-85, essentially contains all the necessary information to the treatment of the case of oriented surfaces, for p-forms.

Supersymmetric structures for second order differential operators

“…As already mentioned, this structure is of great help in the study of the eigenvalue splitting, since in particular it allows one to avoid the study near the so-called "non-resonant wells", such as the saddle point in the example mentioned above. The results were subsequently generalized in [3], [4], in [8] with a full asymptotic expansion, in [14], [15] in cases with boundary, and in [16], [17] in the case of forms of higher degree. Notice that the computation of the exponentially small eigenvalues is performed here using the singular values of the Witten differential.…”

Untitled

“…Similar results have been obtained by Helffer, Klein and Nier in [Helffer et al 2004] using methods from semiclassical analysis. See also [Le Peutrec 2010;2011;Le Peutrec and Nectoux 2021] for generalisations to diffusions on manifolds with or without boundary. The potential-theoretic approach has also been successfully applied to obtain Eyring-Kramers laws for stochastic PDEs Barret 2015;Berglund et al 2017].…”

An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures

“…Another successful approach to sharp asymptotics for metastable transition times is based on semiclassical analysis of the Witten Laplacian, and was initiated in [HKN04,HN05]. Extensions can be found, e.g., in [LP10, LP11,LPNV13].…”