- The hypoelliptic Laplacian, analytic torsion and Cheeger–Müller Theorem. C. R. Acad. Sci. Paris 352 (2014), 153-156. pdf
abstract
The purpose of this Note is to prove a formula relating the hypoelliptic Ray-Singer metric and the Milnor metric on the determinant of the cohomology of a compact Riemannian manifold by a Witten-like deformation of the hypoelliptic Laplacian in de Rham theory.
- Analytic torsion, dynamical zeta functions and orbital integrals. C. R. Acad. Sci. Paris 354 (2016), 433-436. pdf
abstract
The purpose of this Note is to prove an identity between the analytic torsion and the value at zero of a dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold, which solves a conjecture of Fried.
- Laplacien hypoelliptique, torsion analytique, et théorème de Cheeger-Müller. J. Funct. Anal. 270 (2016), 2817-2999. pdf
abstract
L'objet de cet article est de démontrer une formule reliant les métriques de Ray–Singer hypoelliptique et de Milnor sur le déterminant de la cohomologie d'une variété riemannienne compacte par une déformation à la Witten du laplacien hypoelliptique en théorie de de Rham.
- Analytic torsion, dynamical zeta functions, and the Fried conjecture. Analysis & PDE. 11 (2018), 1-74. pdf
abstract
We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of Fried. This article should be read in conjunction with an earlier paper by Moscovici and Stanton.
- Flat vector bundles and analytic torsion on orbifolds. with Jianqing Yu. Preprint. arXiv: 1704.08369.
abstract
This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds.
We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.
- Morse-Smale flow, Milnor metric, and dynamical zeta function. with Jianqing Yu. Preprint. arXiv: 1806.00662.
abstract
With the help of interactions between the fixed points and the closed orbits of a Morse-Smale flow, we introduce a
Milnor metric on the determinant line of the cohomology of the
underlying closed manifold with coefficients in a flat vector
bundle.
This allows us to generalise the notion of the absolute value at zero
point of the
Ruelle dynamical zeta function, even in the case where this
value is not well
defined in the classical sense. We give a
formula relating the Milnor metric and the Ray-Singer metric. An
essential ingredient of our proof is Bismut-Zhang's Theorem.
- Fried conjecture in small dimensions. with N.V. Dang, C. Guillarmou, and G. Rivière.
Preprint. arXiv: 1807.01189.
abstract
We study the twisted Ruelle zeta function \(\zeta_X(s)\) for smooth Anosov vector fields \(X\) acting on flat vector bundles over smooth compact manifolds.
In dimension \(3\), we prove Fried conjecture, relating Reidemeister torsion and \(\zeta_X(0)\). In higher dimensions, we show more generally that \(\zeta_X(0)\) is locally constant with respect to the vector field \(X\) under a spectral condition.
As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic \(3\)-manifolds.
This gives the first examples of non-analytic Anosov flows and geodesic flows in
variable negative curvature where Fried conjecture holds true.