In one paper the author uses the statement without citation:

Let $(M,g)$ be a Riemannian manifold. The gradient $\nabla F$ of a proper function $F: M\rightarrow \mathbb{R}$ is integrable vector field, i.e. its integral curves are defined for all times.

In this particular case, $F$ is also bounded from below, and the Riemannian metric $g$ on $M$ is complete.

My question is

1)Is this statement true in general?

2)If not, why is the statement true when additional assumptions are satisfied (function is bounded from below and metric is complete)?

I could not find a reference on this in the literature; if someone knows it - that will be enough.

Thanks for any help!

  • $\begingroup$ Let $M$ be the open interval $(0,1)$ with its natural metric and $F(x)=x$. Then the gradient flow lines satisfy $\dot{x}=1$ which clearly do not exist for all time. $\endgroup$ – Liviu Nicolaescu Apr 5 '18 at 10:30
  • $\begingroup$ What do you mean by proper? Preimages of compact are compact? $\endgroup$ – Piotr Hajlasz Apr 5 '18 at 11:48
  • $\begingroup$ @LiviuNicolaescu it exists, the flow is $x(t)=x_0+t$. $\endgroup$ – Filip92 Apr 5 '18 at 13:30
  • $\begingroup$ @PiotrHajlasz yes $\endgroup$ – Filip92 Apr 5 '18 at 13:31
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    $\begingroup$ @LiviuNicolaescu $(0,1)$ is not complete and $M$ is assumed to be complete. $\endgroup$ – Piotr Hajlasz Apr 5 '18 at 13:33

Unless there are some additional assumptions imposed, the claim is false. Consider $f(x)=x^4$ on $\mathbb{R}$. This function is bounded from below, $f'(x)=4x^3$, and solutions to $x'(t)=4x(t)^3$ have blowup in a finite time.

  • $\begingroup$ I suspect that there is a sign difference, and the gradient flow of F is understood to be the flow in the direction of $-\nabla F$. $\endgroup$ – Michael Renardy Apr 5 '18 at 12:42
  • $\begingroup$ @Piotr Hajlasz: This is true indeed, thank you. So, one might impose an additional condition that $(M,g,I)$ is Kahler, and $F$ is a moment map of an I-holomorphic, isometric $\mathbb{S}^1-$action. This is the setup from the paper I mentioned. $\endgroup$ – Filip92 Apr 5 '18 at 13:39

Theorem. Suppose $(M,g)$ is a Riemann manifold $\newcommand{\bR}{\mathbb{R}}$ and $f:M\to\bR$ is a a smooth function such that the sublevel sets $\{f\leq c\}$ are compact for any $c\in\bR$. Then the negative gradient flow exists for all positive times.

Proof. Take a regular value $c$. By Sard's theorem most $c$ are regular values. The sublevel set $\{f\leq c\}$ is a compact manifold with boundary and the vector field $-\nabla f$ is perpendicular to the boundary $\{f=c\}$ and points towards the interior because $f$ decreases along the negative gradient flow.


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