Question

Suppose that if $X$ is a complete, simply connected Kaehler manifold with non-positive sectional curvatures. Let $P \in X$ and $h : X \to \mathbb{R}$ be the function defined by $h(x) = dist(P,X)^2$. Is it true that the Levi form $(\partial^2 h/\partial z_j\partial \bar{z}_k)(x)$ is positive definite at each point $x \in X$, $x\neq P$? If not, is it true when $X$ is a hermitian symmetric domain? Or when $X$ is the Siegel upper half space of rank $g$?

Answer 1

By the Hessian comparison theorem the square of the distance function on X is strictly convex. On Kahler manifolds strictly convex functions are strictly plurisubharmonic .By Grauert's solution of the Levi problem X is Stein.See R E Greene and H H Wu Springer LNM 699 . There is an example of a complete simply connected negatively curved Hermitian manifold which is not Stein due to P Klembeck.So we need the Kahler assumption.