Riemannian manifolds with entire Grauert tube are rationally elliptic

TL;DR

This paper proves that compact simply connected real analytic Riemannian manifolds with entire Grauert tubes are rationally elliptic, confirming a conjecture for a specific class of manifolds with nonnegative sectional curvature.

Contribution

It establishes the rational ellipticity of manifolds with entire Grauert tubes, advancing the understanding of their topological and geometric properties.

Findings

Confirmed the Bott-Grove-Halperin conjecture for manifolds with entire Grauert tubes

Demonstrated that such manifolds are rationally elliptic

Provided new insights into the structure of real analytic Riemannian manifolds

Abstract

It was conjectured by Bott-Grove-Halperin that a compact simply connected Riemannian manifold $M$ with nonnegative sectional curvature is rationally elliptic. We confirm this conjecture under the stronger assumption that $M$ has entire Grauert tube,i.e.,$M$ is a real analytic Riemannian manifold that has a unique adapted complex structure defined on the whole tangent bundle $TM$.