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During the last decades, several investigations were concerned with rigidity statements for manifolds without conjugate points (some results can be found in the references). Based on an idea by E. Hopf, K. Burns and G. Knieper proved that cylinders without conjugate points and with a lower sectional curvature bound must be flat if the length of the shortest loop at every point is globally bounded.
The present article reduces the last condition to a limit for the asymptotic growth of loop-length as the basepoint approaches the ends of the cylinder (Thm. 18). Along the way, the shape of cylinders without conjugate points is characterized: The loop-length must be strictly monotone increasing to both ends outside a – possibly empty – tube consisting of closed geodesics (Thm. 10).
We study in this note the Dirichlet problem for complex Monge-Ampère equation in compact Stein manifolds with boundary. As far as we know among the global results for Monge- Ampère equations, compact manifolds with boundary have been less discussed.
We show that if M = X × Y is the product of two complex manifolds (of positive dimensions), then M does not admit any complete Kähler metric with bisectional curvature bounded between two negative constants. More generally, a locally-trivial holomorphic fibre-bundle does not admit such a metric.