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Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and don't depend on the metric. But its gradient vector field does depend on the metric.

Now we would like to change the metric to be $g_\epsilon:= g_M\oplus \epsilon g_N$, with $0<\epsilon< \infty$, and consider the process $\epsilon\to \infty$. Intuitively, the $N$-component of the gradient will be very small. Consider two critical points $(p_i, q_i)$, $i=1, 2$ and the moduli space ${\mathcal M}_\epsilon$ of trajectories between $(p_i,q_i)$ for the metric $g_\epsilon$. So it seems that objects in the limit moduli would be the union of a bunch of trajectories on $M$, of functions $f(\cdot, y_k)$, with several different $y_k$'s. Is that necessary that those trajectories(in $M$) connect successively? Are there other types of limit objects?

More precisely, suppose $\epsilon_j\to \infty$, and $\gamma_j\in {\mathcal M}_{\epsilon_j}$. What are all the possible limits(in a proper sense) of subsequences of $\gamma_j$? If we know all the limits, can we expect a gluing argument to add a good end of the universal moduli $\cup_\epsilon {\mathcal M}_\epsilon$ on the $\epsilon=\infty $ side?

We may assume that the function and the metrics $g_M$, $g_N$ are all generic.

This question can be asked more generally: if we have a distribution on a manifold(or a foliation), and we shrink the metric on the direction perpendicular to the foliation, what do the limit Morse trajectories look like?

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up vote 4 down vote accepted

Don't you want to take $\epsilon\to\infty$ to make the $N$ component of the gradient small?

Anyway, here is roughly what I think happens. For each $y\in N$ there is a function $f(\cdot,y)$ on $M\times\{y\}$ with some critical points. The unions of these critical points form submanifolds (maybe with singularities, but for simplicity let us assume that these do not exist) of $M\times N$. Let's call these "critical submanifolds". Now $f$ restricts to a function $f_S$ on each critical submanifold $S$, which with luck is Morse. The critical points of the functions $f_S$ on the different critical submanifolds $S$ are exactly the critical points of the original function $f$ on $M\times N$. Now in the limit as $\epsilon\to\infty$, I think that a gradient flow line degenerates to the following type of object: You start at a critical point of some $f_S$, follow the gradient flow of $f_S$ to some noncritical point $(x,y)\in S$, then follow the gradient flow of $f(\cdot,y)$ to a point $(x',y)$ in some other critical sumbanifold $S'$, then continue along $S'$, and so on, eventually stopping at a critical point of the function on some other critical submanifold. This is similar to the "cascade" picture of Morse-Bott theory. I have thought for a while that in the special case when $N$ is an interval, you can use this picture to show that continuation maps (giving the isomorphism between the Morse homologies of two different Morse-Smale pairs on $M$) defined in the usual Floer-theoretic way agree with the isomorphisms you can write down by hand from studying how the Morse complex changes as a result of bifurcations.

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Yes, should be $\epsilon\to \infty$. –  Guangbo Xu Aug 10 '11 at 13:11
    
@Michael: I am trying to write down the details. One problem is that, on the critical submanifold $S$, there might be a codimension 1 submanifold which corresponds to those DEGENERATE critical points of $f(\cdot, y)$. When crossing these walls, the Morse index may jump. So when we try to construct the limit object for a given sequence, we try to find trajectories of $f(\cdot, y)$ from a component of $S$ to another component(by induction), and hope the induction may stop at finite times. But because of the jumping of indices, I have some trouble on deducing the finiteness. –  Guangbo Xu Aug 14 '11 at 18:11
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