It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I suspect that the converse doesn't hold, though I haven't tried all that hard to find a counterexample. I am curious about what kinds of additional assumptions one would need to make in order for something like the converse to hold. That is:

I'd like to know when, given a Stein manifold $X$, we are able to deduce the existence of a complete metric of nonpositive sectional curvature on $X$ (I don't need the metric to be Kaehler). In case it helps, I am especially interested in the case where $X$ is a subvariety of a hermitian symmetric domain of noncompact type.

I have some thoughts on this. For example, let's start with the observations that every Stein manifold can be embedded in $\mathbb{C}^N$ for some $N$, and that holomorphic sectional curvature decreases on complex subvarieties. Then the Euclidean metric on $\mathbb{C}^N$ induces a complete metric of nonpositive holomorphic sectional curvature on a Stein manifold $X\subset \mathbb{C}^N$. The wrinkle here is that nonpositive holomorphic sectional curvature is weaker than nonpositive sectional curvature.

It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I am curious about what kinds of additional assumptions one would need to make in order for something like the converse to hold. That is:

I'd like to know when, given a Stein manifold $X$, we are able to deduce the existence of a complete metric of nonpositive sectional curvature on $X$ (I don't need the metric to be Kaehler). In case it helps, I am especially interested in the case where $X$ is a subvariety of a hermitian symmetric domain of noncompact type.

I have some thoughts on this. For example, let's start with the observations that every Stein manifold can be embedded in $\mathbb{C}^N$ for some $N$, and that holomorphic sectional curvature decreases on complex subvarieties. Then the Euclidean metric on $\mathbb{C}^N$ induces a complete metric of nonpositive holomorphic sectional curvature on a Stein manifold $X\subset \mathbb{C}^N$. The wrinkle here is that nonpositive holomorphic sectional curvature is weaker than nonpositive sectional curvature.

It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I suspect that the converse doesn't hold, though I haven't tried all that hard to find a counterexample. I am curious about what kinds of additional assumptions one would need to make in order for something like the converse to hold. That is:

I'd like to know when, given a simply-connected Stein manifold $X$, we are able to deduce the existence of a complete metric of nonpositive sectional curvature on $X$ (I don't need the metric to be Kaehler). In case it helps, I am especially interested in the case where $X$ is a subvariety of a hermitian symmetric domain of noncompact type.

I have some thoughts on this. For example, let's start with the observations that every Stein manifold can be embedded in $\mathbb{C}^N$ for some $N$, and that holomorphic sectional curvature decreases on complex subvarieties. Then the Euclidean metric on $\mathbb{C}^N$ induces a complete metric of nonpositive holomorphic sectional curvature on a Stein manifold $X\subset \mathbb{C}^N$. The wrinkle here is that nonpositive holomorphic sectional curvature is weaker than nonpositive sectional curvature. I know I haven't yet used the hypothesis of simple-connectedness, but I don't really see where that would come in.

It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I suspect that the converse doesn't hold, though I haven't tried all that hard to find a counterexample. I am curious about what kinds of additional assumptions one would need to make in order for something like the converse to hold. That is:

I'd like to know when, given a Stein manifold $X$, we are able to deduce the existence of a complete metric of nonpositive sectional curvature on $X$ (I don't need the metric to be Kaehler). In case it helps, I am especially interested in the case where $X$ is a subvariety of a hermitian symmetric domain of noncompact type.

I have some thoughts on this. For example, let's start with the observations that every Stein manifold can be embedded in $\mathbb{C}^N$ for some $N$, and that holomorphic sectional curvature decreases on complex subvarieties. Then the Euclidean metric on $\mathbb{C}^N$ induces a complete metric of nonpositive holomorphic sectional curvature on a Stein manifold $X\subset \mathbb{C}^N$. The wrinkle here is that nonpositive holomorphic sectional curvature is weaker than nonpositive sectional curvature.