1
$\begingroup$

For an open subset $U$ of $\mathbb{C}^{2}$ containing $0$ and a holomorphic map $f:U\to \mathbb{C}^{2}$ which has a unique zero at the origin we associate a natural singular holomorphic foliation by complex curves, the foliation arising from $\dot z=f(z)$. In this case origin is called a singularity.

What is an example of such singular foliation for which the foliation is stable at $0$?

That is, for every open set $W$ containing origin, there is an open set $V\subset W$ which saturation under foliation is contained in $W$? Is there an indirect relation between this question and some methods in "Minimal set problem" about singular holomorphic foliations of $\mathbb{C}P^{2}$?

Note that a minimal set for a singular foliation of $\mathbb{C}P^{2}$ is a compact subset which is invariant(saturated) under foliation and does not contain any singular point. For "minimal set" see here.

$\endgroup$
3
+50
$\begingroup$

Such an example is impossible. We can always assume $W$ is a polydisc, and part of its boundary $\partial W$ is included in the $3$-space $T=\{(x,y) : |y|=r\}$. Take $p\in T\cap\bar W$. If the foliation were stable then the image of $t\mapsto z(t))$, for small $t$ and with $z(0)=p$, would be included in the adherence of a single connected component of $\mathbb C^2\setminus T$. This contradicts the maximum/minimum modulus principle for the $y$-component of $z(t)$ except if $z=cst$, but in that case the trajectory escapes from $W$ through another component of the boundary. Therefore every leaf of the foliation passing through $\partial W$ escapes from $W$.

You can adapt this construction to the saturation $S$ of $V$ in $W$, to prove the foliation always reaches $\partial W$. If it were not the case, consider the smallest closed polydisc $\bar W_S\subsetneq \bar W$ containing $S$. There is a point $p\in\partial S \cap \partial W_S$, to which the above argument applies: either the trajectory passing through $p$ escapes from $W_S$ (contradiction) or it is included in $\partial W_S$ and therefore leaves the polydisc through another component of $\partial W_S$ (contradiction again).

$\endgroup$
10
  • $\begingroup$ In fact every non-singular trajectory escapes from $W$. $\endgroup$ May 11 '15 at 16:07
  • $\begingroup$ Thanks for your answer. I think I can not understand some thing in the last part of your argument or some thing is missing in your answer. An equivalent definition of stability is as follows :for every open set $W$ there is an open set $V\subset W$ such that $V$ is saturated. But why we can choose $V$ in the form of a poly disc? In fact the last part of your statement (which does not depend on real or complex case, I think) is not valid in real case (the center). Moreover By 3-torus do you mean solid two torus?(The boundary is the union of two solid torus) $\endgroup$ May 11 '15 at 16:52
  • $\begingroup$ Moreover what is the contradiction if in every small neighborhood we have a leaf with constant y-coordinate? Note that we do not assume that the singularity is non degenerate.May be it has vanishing linear term. $\endgroup$ May 11 '15 at 17:14
  • $\begingroup$ Or vanishing k-Jet for k large. $\endgroup$ May 11 '15 at 17:26
  • 1
    $\begingroup$ @AliTaghavi:I never claimed you could take $V$ as a polydisc, I'm sorry if I was unclear. I modified the post in a way which is clearer I hope. Also, there is no problem in having trajectories with constant $y$-coordinate, they just escape from somewhere else. $\endgroup$ May 11 '15 at 18:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.