Vortices over Riemann surfaces and dominated splittings

Abstract

We associate a flow $\varphi$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\varphi$ always admits a dominated splitting and identify special cases in which $\varphi$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.

Exposition improved, subsection 3.5 added, typos corrected, references updated, 27 pages

References in corpus (5)

• Anosov representations and dominated splittings
• A (short) survey on Dominated Splitting
• Convex projective surfaces with compatible Weyl connection are hyperbolic
• Holomorphic differentials, thermostats and Anosov flows
• Minimal Lagrangian connections on compact surfaces

Cited by in corpus (2)

• Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle
• Smooth orbit equivalence rigidity for dissipative geodesic flows