Zero entropy on entire Grauert tubes

TL;DR

This paper proves that manifolds with entire Grauert tubes have zero topological entropy in their geodesic flow, leading to new classifications and restrictions on the topology of such manifolds.

Contribution

It establishes the zero entropy property for entire Grauert tubes and provides comprehensive classifications and topological restrictions for manifolds with this property.

Findings

Geodesic flow of manifolds with entire Grauert tubes has zero topological entropy.

Grauert tubes of convex analytic hypersurfaces in R^3 are generically finite.

Classifies 3-manifolds with entire Grauert tubes as those admitting good complexifications.

Abstract

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.