Given a Stein Manifold $Y$, I can consider a local Vector field around a point $p\in K$, where $K\subset Y$ is a compact; e.g. in local coordinates, it can be a constant vector field. In local coordinates I can write $V(z)=v$, on $U$ a neighborhood of $p$. This $V$ can be obviously extended to the whole $\Bbb C^n$: this is because tangent spaces in $\Bbb C^n$ are basically just $\Bbb C^n$, while in a manifold $T_pK$ changes as $p$ changes. So in $\Bbb C^n$ I would get a global flow.

My problem is that this doesn't hold true in a manifold anymore: I would only have a local flow near $p$. I suspect the flow can be holomorphically extended on a whole neighborhood of $K$, but I cannot find any argument for this. Any hint? Thanks

Given a Stein manifold $Y$, I can consider a local vector field around a point $p\in K$, where $K\subset Y$ is compact; e.g. in local coordinates, it can be a constant vector field. In local coordinates I can write $V(z)=v$, on $U$ a neighborhood of $p$. This $V$ can be obviously extended to the whole $\Bbb C^n$: this is because tangent spaces in $\Bbb C^n$ are basically just $\Bbb C^n$, while in a manifold $T_pK$ changes as $p$ changes. So in $\Bbb C^n$ I would get a global flow.

My problem is that this doesn't hold true in a manifold anymore: I would only have a local flow near $p$. I suspect the flow can be holomorphically extended on a whole neighborhood of $K$, but I cannot find any argument for this. Any hint? Thanks