1
$\begingroup$

I am trying to understand a passage in Gromov paper "Singularities, expanders and the topology of maps", p. 12 (Section 2.1).

At some point he constructs a "generic" non-vanishing non-exact 1-form $\varphi$ in $\mathbb R^2$ such that $\varphi = d y_1$ at infinity ($y_1$, $y_2$ are the standard coordinates of $\mathbb R^2$) with some properties that maybe relevant to this question or not.

Then he invokes the poincaré-bendixson theorem, claiming that it provides a smooth non-vanishing function $\rho \: \mathbb R^2 \to \mathbb R$ such that $\rho \varphi$ is exact.

Do you have any idea about how that could work? is there some more suited version of the poincaré-bendixson theorem involved?

  • 0
    Just a *minor* comment on what it looks like (as most of the time, Gromov kills it instantly): Looks like he is constructing a gradient flow starting with the equilibria and "circles", instead of going in the other direction. The nonexistence of equilibria in some regions should ensure that the gradient flow has the circles has periodic orbits. – 2017-01-27
  • 0
    Hi! did you mean: "has the circles AS periodic orbits?". But how can a gradient flow have periodic orbits at all? Maybe he wants to prove that the gradient flow does not admit periodic orbits. And maybe this is a more or less sufficient condition for a flow to be a gradient flow.. – 2017-01-27
  • 0
    Yes, sorry, "as". Often I find it difficult to follow Gromov. It only seems to me that it should be something along this direction, having in mind what the PB theorem says. – 2017-01-27

0 Answers 0