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Let $(M, g)$ be a compact Riemannian manifold and $f: M \rightarrow \mathbb{R}$ be a Morse-Bott function, i.e. the set a critical points of $f$, $Crit(f)$, has connected components which are smooth manifolds and which have as tangent spaces $T_x Crit(f) = \ker \nabla^2_x f$,

(where $\nabla^2_x f: T_x M \rightarrow T_xM$ is the linear operator obtained via $g$ from the hessian $f_{**,x} : T_x M \times T_x M \rightarrow \mathbb{R}$ defined as $f_{**,x}(v, w) = v(W(f))$ for $W \in \Gamma(TM)$ any extension of $w$ (this is well defined and symmetric at critical points) )

Let $\nabla f \in \Gamma(TM)$ be defined by $g(\nabla f, w) = w(f)$ and consider the flow of $-\nabla f$ denoted $\phi_t(y)$. I am trying to see why for any $y \in M$ it happens that $\lim\limits_{t \rightarrow \infty} \phi_t(y) \in Crit(f)$.

My attempt:

Since $M$ is compact, $\phi_t(y)$ is defined for all $t \in \mathbb{R}$. If the set $A_y:= \{ \phi_t(y) : t \in \mathbb{R} \}$ were closed, then, since $M$ is compact, this set would also be compact, and by Weierstrass $f$ would have to attain its minimum on it. Since moving along the flowlines of the negative gradient can only decrease $f$, this means that the minimum is attained at $x:= \lim\limits_{t \rightarrow \infty} \phi_t(y)$, so then $x$ would be a critical value for $f|_{A_y}$. But even ignoring the fact that I don't know why $A_y$ is necessarily closed, I don't see why if $x$ is a critical value for $f|_{A_y}$, then it is a critical value for $f$ as well.

I am thinking that this attempt not enough, as it doesn't use at all the fact that $f$ is a Morse-Bott function. But I don't see how to use this fact. I also know that $\nabla_x^2 f (v) = \nabla_V \nabla f$ for $x \in Crit(f)$ and $v \in T_xM$ and $V$ a vector field extending $v$, where $\nabla_V (\cdot)$ in the RHS is the Levi-Civita connexion of $g$, but I can't see how to use this either.

rosecabbage
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    You have to use the Morse Bott assumption. There are cases where the negative gradient flow does not converge. – Arctic Char Oct 02 '21 at 12:52
  • I realised I haven't used the Morse-Bott assumption and added an extra paragraph at the end. Could you explain or at least give a hint on how to use this assumption to show convergence, and moreover convergence to a critical point? – rosecabbage Oct 02 '21 at 12:57
  • Do you know how to do it when $f$ is a morse function? – Arctic Char Oct 02 '21 at 13:07
  • @ArcticChar No. – rosecabbage Oct 02 '21 at 13:15
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    Please can someone give the proof that The negative gradient flow of Morse-Bott function converges to a critical point ? – Mira Jun 25 '22 at 09:10
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    @Asma thank you for setting the bounty. This question still puzzles me after a year. There is some theory done in Banyaga and Hurtubise, Lectures on Morse Homology, chapter 4.2, but it's a highly technical, difficult to read chapter (they even say it contains the most difficult real analysis in the book) and it's done in the general setting of Banach spaces. I still don't know if there are simpler proofs for manifolds. – rosecabbage Jun 25 '22 at 09:26
  • CrossPosted: https://mathoverflow.net/questions/425477/the-negative-gradient-flow-of-a-morse-bott-function-on-a-compact-manifold-conver – Mira Jun 25 '22 at 23:11
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    Generally speaking $A_y$ will not be closed. You ought to think through an example when cooking up an argument like this. For example, take a linear height function on a circle or sphere. The only flowlines that are closed are critical points. So you have two tasks, 1) show the limit exists, and 2) that it is a critical point. – Ryan Budney Jun 26 '22 at 07:54
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    Perhaps the problem is you are maybe trying to prove the wrong theorem. i.e. do you really need the Morse-Bott condition? i.e. argue if you flow along a gradient curve for any smooth function $f : M \to \mathbb R$, provided $M$ is compact, then the limit of the flowline exists and is a critical point of $f$. For existence, notice that the function $f$ is monotone on a flowline. – Ryan Budney Jun 26 '22 at 07:58
  • This question is answered here https://mathoverflow.net/questions/425477/the-negative-gradient-flow-of-a-morse-bott-function-on-a-compact-manifold-conver/429039#429039 – Mira Aug 25 '22 at 02:17

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