Morse theory and adiabatic limits - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:12:56Z http://mathoverflow.net/feeds/question/72527 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72527/morse-theory-and-adiabatic-limits Morse theory and adiabatic limits Guangbo Xu 2011-08-09T23:13:15Z 2011-08-10T13:12:28Z <p>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. </p> <p>Now we would like to change the metric to be $g_\epsilon:= g_M\oplus \epsilon g_N$, with $0&lt;\epsilon&lt; \infty$, 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?</p> <p>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? </p> <p>We may assume that the function and the metrics $g_M$, $g_N$ are all generic.</p> <p>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?</p> http://mathoverflow.net/questions/72527/morse-theory-and-adiabatic-limits/72553#72553 Answer by Michael Hutchings for Morse theory and adiabatic limits Michael Hutchings 2011-08-10T05:54:23Z 2011-08-10T05:54:23Z <p>Don't you want to take $\epsilon\to\infty$ to make the $N$ component of the gradient small?</p> <p>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.</p>