  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 $TM$.
