Orbits of the geodesic flow and chains on the boundary of a Grauert tube

Abstract

A \(C^{\omega}\) Riemannian manifold (X,g) determines an integrable complex structure on a tubular neighborhood, \(T^{*\epsilon}X\), of the zero section in \(T^{*}X\) and a CR-structure on the boundary, \(M^{\epsilon}\). There are two natural families of curves on \(M^{\epsilon}\): the orbits of the geodesic flow and a CR-invariant family called chains. It is natural to ask whether they are related. We show that if orbits of the geodesic flow are chains on \(M^{\epsilon}\) for all \(\epsilon\) sufficiently small, then (X,g) is Einstein. As a partial converse we show that if (X,g) is harmonic, then orbits of the geodesic flow are chains. To prove this we study the Fefferman metric associated with \(M^{\epsilon}\).