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I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded.

To be more precise, a Morse-Bott function on a Riemannian manifold $(M,g)$ is a smooth map $f:M \longrightarrow \mathbb R$ such that the set of critical points $Crit f \subset M$ is a submanifold and $T_x Crit f = ker Hess f_x$ for all $x \in Crit f$. It might be more convenient to express this by using the linear map $A_x:T_xM \longrightarrow T_xM$ given by $g_x(A_xv,w) = Hess f_x(v,w)$ for $v,w \in T_xM$. I am interested in the behaviour of maps $x:\mathbb R_+ \longrightarrow M$ satisfying $\dot x(t) = -\nabla f(x(t))$. Does $lim_{t \to \infty} x(t)$ exist?

An easy computation shows that $df_{x(s)} \longrightarrow 0$ for $s \to \infty$. So if $x$ is bounded (e.g. if $M$ is compact) $x$ has to "converge" towards $Crit f$. UNfortunately, this does not rule out that $x$ has several limits in $Crit f$. (This can not happen for Morse functions, since in this case the critical points are isolated).

  1. The simplest case is the case where $M$ is finite dimensional. As far as I can see, the answer is already given by Austin and Braam in their paper "Morse-Bott theory and equivariant cohomology", 1995. In particular Theorem A.9 is important, stating that the stable manifold of a connected component $C \subset Crit f$, given by $W^s(C) = \{x \in M | \phi_t(x) \to C\}$ where $\phi$ is the flow of the negative gradient, is indeed a submanifold.

  2. The second case is the case where $M$ is a Hilbert manifold (infinite dimensional) endowed with a complete metric on the tangent bundle. Although I could only find a paper dealing with this question in the Morse case (see Abbondandolo, Majer: "A Morse complex for infinite dimensional manifolds", Appendix C), I assume that it should be not to hard to extend the result to the Morse-Bott case. The important point here is that the operator $A_x$ mentioned above is symmetric and defined on the whole tangent space, so it has to be bounded.

  3. The last case is the one I am particularly interested in. We start again with a Hilbert manifold $M$, but now endowed with an incomplete Riemannian metric (in my case, I have a Sobolev space $W^{1,2}$ which is only endowed with the $L^2$ metric). Let $H$ denote the complete Hilbert space where $T_xM \subset H$ is dense. Then the operator $A$ is unfortunately of the form $A_x:T_xM \subset H \longrightarrow H$. Now $A_x$ is unbounded, but at least still symmetric and even essentially self-adjoint. However, here I am failing to apply the results from case 2, which strongly used the boundedness of $A$. I would be very content if a similar result could be shown at least for the case, where $Crit f$ is finite dimensional.

A remark: In the infinite dimensional case, one might have problems with the flow, since it might not be defined for all times. However, I start with a flow line which is already defined for all $t$. Furthermore, I have found (with completely different techniques) a sequence $t_k \to \infty$ such that $x(t_k) \to p \in Crit f$, which is enough to ensure that $x(t) \to Crit f$.

I would also be thankful for counterexamples where the gradient flow does not have a unique limit. But I expect those examples not to contain Morse-Bott functions.

Schlubbidubbi
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    My first problem is that gradient flows even of functions with bounded derivative may want to leave the manifold if you are not in the complete setting. You say that this does not take place, is this an extra assumption you are making or a feature of the specific thing you are looking at? – s.harp Jan 05 '19 at 13:48
  • It seems to me that the definition you provide for a Morse-Bott function is not accurate. – Amitai Yuval Jan 05 '19 at 14:43
  • It should be enough to assume that the flow line is bounded (the flow line itself, not the derivative). And indeed, the existence of the sequence on which the flow line already converges is an extra assumption which holds in my specific setting. And of course there is a typo in the definition of a Morse-Bott function which I just corrected – Schlubbidubbi Jan 05 '19 at 17:13
  • Could this be related to your question? – Evgeny Jan 05 '19 at 17:22
  • It certainly is. It is an example where the flow line on a compact finite dimensional manifold does not converge. But although I have not checked it properly I stongly assume that the function is not Morse-Bott. The case of Hilbert manifolds is unfortunately not menrtioned. – Schlubbidubbi Jan 05 '19 at 17:32
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    I am certain your question is (at least essentially) correct for compact component $C \subset \text{Crit}(f)$. You would write down the gradient flow equation in a neighborhood of $C$ in terms of the Hessian and some small higher order terms. Nondegeneracy of the Hessian should be used to argue that the part "normal to $C$", if it converges to 0, converges rather quickly, use this to bound the non-linear term, and see that the nonlinear term governs movement in $C$. However, the details of this approach are technical enough I am not really willing to spend time writing it up, unfortunately. –  Jan 06 '19 at 13:41
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    For analytic functions which are not necessarily Morse-Bott (I know this isn't your question!) I think everyone working in this area should be aware of Leon Simon's version of the Lojasewicz inequality; I first learned an exposition of this in a book by Morgan-Mrowka-Rubetman. –  Jan 06 '19 at 13:43

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