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).
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.
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.
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 am only interested in the case where the flow line really "converges" towards $Crit f$ since in my set up, this has already been proven with completely different methods.
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.
