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 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.