Riemannian manifolds with entire Grauert tube are rationally elliptic

It was conjectured by Bott, Grove and 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, ie M is a real-analytic Riemannian manifold that has a unique adapted complex structure defined on the whole tangent bundle TM .Our result also provides a strong topological obstruction to the existence of an entire Grauert tube.53C20