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Harmonic 1-forms on the stable foliation. (English) Zbl 0821.58034

Author’s introduction: Let \((M,g)\) be a closed Riemannian manifold with negative sectional curvature, let \(W^ s\) denote the stable foliation of the unit tangent bundle \(SM\), and consider the differential complex of sections of real \(p\)-forms on \(TW\) which are smooth along the leaves of the foliation and such that all jets are globally Hölder continuous. The cohomology is trivial except in degree one where a closed 1-form \(\alpha\) is exact if and only if \(\int_ \gamma \alpha = 0\) for every closed curve \(\gamma\) which remains in the same leaf.
In this paper are studied some properties of cohomology classes of closed 1-forms. Our main result is that in each cohomology class there is a unique harmonic 1-form. We also define an “asymptotic cycle” and an “asymptotic energy” on cohomology classes, naturally associated with the leafwise Laplacian. In fact these properties are related to asymptotic properties of the leafwise heat kernel, which were established by the author in ‘Central limit theorem in negative curvature’ (Preprint). Using the idea of Y. Le Jan [The central limit theorem for the geodesic flow on non-compact manifolds of constant negative curvature (to appear in Duke Math. J.)], we are able to express the above asymptotic quantities through the thermodynamical formalism of the geodesic flow.
Reviewer: D.Hurley (Cork)

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37D99 Dynamical systems with hyperbolic behavior
31C12 Potential theory on Riemannian manifolds and other spaces
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