Some complete invariant metrics in Grauert tubes

We first show the existence of the complete invariant \K metric $-\log (r^2-\rho)$ on the \gt\ $T^rX$ over \h $X$. The existence of a complete \ke\ of negative Ricci curvature on $T^rX$, when $X$ is semi-compact, is also proved. If $X$ is of rank-one symmetric then near the center there exists a \K potential depending solely on the \ma solution $\sqrt\rho$ and the solving of the complete \ke\ is reduced to the solving of an ODE. The restriction to the center of the complete \ke\ is proportional to the original Riemannian metric of $X$ is proved. The holomorphic sectional curvatures of this \ke\ along leaves of the \ma foliation near the center is estimated.

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Published 1 January 2007