Victor Donnay (Bryn Mawr)

Embedded surfaces with Anosov geodesic flow.

Abstract. We construct surfaces, isometrically embeddable in R³, whose geodesic flow is Anosov. It is known that no metric on surfaces of genus 0 or 1 can have an Anosov geodesic flow, while for surfaces of genus 2 or higher there exist metrics of negative curvature that produce Anosov flows, but these negatively curved metrics can not be isometrically embedded.

Our construction produces surfaces with Anosov geodesic flow for all sufficiently large genus, but does not produce any explicit bounds on the genus. An open question is whether for all genus above 1, there exist embeddable metrics with Anosov geodesic flow or, if not, to give lower bounds on the genus. This result is joint work by V.J. Donnay and C. Pugh and builds on their earlier work of constructing embedded surfaces with Bernoulli geodesic flow using the finite horizon cap construction.