Given a Stein manifold $Y$, there exists $\rho$, a $\mathscr C^2$-smooth strictly plurisubharmonic exhausting function for $Y$, such that the set of critical points $C=\{z\in Y\;:\;d\rho(z)=0\}$ is discrete; calling $Y_c:=\{\rho\le c\}$, we have that $\bigcup_{c\in\Bbb R}Y_c=Y$.
Consider a compact $K\subset Y$; then there exists $c\in \Bbb R$ such that $K\subset Y_c$ and $K\cap \partial Y_c\neq\emptyset$. But how can I guarantee that in such nonempty intersection I can find a non singular point?
Let us say that $K\cap \partial Y_c$ is just one point $p$ and it stays in $C$.
Since $C$ is discrete, my thought is to consider a vector field $V\colon Y\to Y$ and its flow $\phi_t$ on $K$ for sufficiently small times $|t|<<1$; so $\phi_t(K)$ is an arbitrarily little perturbation of $K$. Nevertheless, this does not guarantee to move the point $p$. How can I do it?
In general, given a compact $K$, how can I find regular points on the boundary of $K$ belonging to $\partial Y_c$ for some $c$? An hint on where to look at will be ok. Thanks.
