Zero entropy on entire Grauert tubes

On a real analytic Riemannian manifold a Grauert tube is an uniquely adapted complex structure defined on the tangent bundle. It is called entire if it may be defined on the whole tangent bundle. Here, we show that the geodesic flow of an analytic manifold with entire Grauert tube has zero topological entropy. Several consequences then follow. We find that Grauert tubes of convex analytic hypersurfaces of ${\bf R}^3$ are generically finite. We give a complete classification of 3-manifolds with entire Grauert tube, showing that these are precisely the 3-manifolds that admit good complexifications. Assuming a manifold has entire Grauert tube, we offer classification statements for several families of smooth 4-manifolds, determine all such simply connected 5-manifolds, and offer new topological restrictions for certain manifolds with infinite fundamental groups. Using these results we present an example of a simply connected and rationally elliptic 5-manifold which nonetheless does not admit any metric with entire Grauert tube.