Surgery on Anosov flows using bi-contact geometry

TL;DR

This paper introduces a novel Dehn surgery technique on Anosov flows via bi-contact geometry, linking contact structures and Reeb dynamics to characterize when such flows remain Anosov after surgery.

Contribution

It defines a new type of surgery on Anosov flows using bi-contact geometry and establishes conditions for resulting flows to be contact Anosov flows, especially via Legendrian surgery on geodesic flows.

Findings

New Dehn surgery method for Anosov flows

Characterization of when surgeries produce contact Anosov flows

Conditions for Legendrian surgery on geodesic flows to yield Anosov flows

Abstract

Using bi-contact geometry, we define a new type of Dehn surgery on an Anosov flow with orientable weak invariant foliations. The Anosovity of the new flow is strictly connected to contact geometry and the Reeb dynamics of the defining bi-contact structure. This approach gives new insights into the properties of the flows produced by Goodman surgery and clarifies under which conditions Goodman's construction yields an Anosov flow. Our main application gives a necessary and sufficient condition to generate a contact Anosov flow by Foulon-Hasselblatt Legendrian surgery on a geodesic flow. In particular we show that this is possible if and only if the surgery is performed along a simple closed geodesic. As a corollary we have that any (positive) skewed R-covered Anosov flow obtained by surgery on a closed orbit of a geodesic flow is orbit equivalent to a (positive) contact Anosov flow.