Abstract
We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian two-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.
Highlights
We introduce a new family of flows on the unit tangent bundle S M of a closed oriented Riemannian two-manifold (M, g) of negative Euler characteristic
The flows are thermostat flows and are generated by C∞ vector fields of the form F := X + (a − V θ )V, where X, V denote the geodesic and vertical vector fields on S M, θ is a 1-form on M—thought of as a real-valued function on S M—and a represents a differential A of degree m 2 on M
We show that a triple (g, A, θ ) satisfying the Eq (1.1) determines a holomorphic line bundle structure on the smooth complex line bundle Lm := 2(T M)(m−1)/2 ⊗ C, so that the “weighted differential” P =−(m−1)/4 ⊗ A is a holomorphic section of and such that a certain negative curvature condition holds
Summary
We introduce a new family of flows on the unit tangent bundle S M of a closed oriented Riemannian two-manifold (M, g) of negative Euler characteristic. We show that the case where A vanishes identically corresponds to W-flows associated to conformal connections on the tangent bundle of a surface that have negative definite symmetrised Ricci curvature. We recover [41, Theorem 5.2], by showing that the flow associated to a triple (g, 0, θ ) solving (1.1) is Anosov This is achieved by providing sufficiency conditions for a general thermostat flow to admit a dominated splitting and to have the Anosov property, see Proposition 3.5 and Theorem 3.7. Since the Anosov property is invariant under reparametrisation, we may ask if the thermostat flow associated to a pair (g, A) solving (1.2) is Anosov for all m 2 This is the case, we obtain: Theorem 5.1 Let (g, A) be a pair satisfying the coupled vortex equations ∂ ̄ A = 0 and Kg = −1 + (m − 1)| A|2g.
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