Morse Functions Exist

The astute reader at this point may be getting a little anxious that despite the fact that I found Morse function in two easy low dimensional cases, my eventual goal of saying very general things about manifolds by using Morse functions is going to rely on the fact that they exist.

If these thing are really as powerful as I have been making them out to be, then it would seem that there probably isn’t an abundance of them. But surprisingly, it turns out that basically every smooth function is Morse.

Let M^n be a closed manifold, and g:M\to \mathbb{R} be a smooth function. Then there is a Morse function f:M\to\mathbb{R} arbitrarily close to g.

Recall Sard’s Theorem (I’m assuming some familiarity with it, which is probably not a good idea): The set of critical values of a smooth map f: U\to \mathbb{R}^n has measure zero in \mathbb{R}^n.

Now we’ll first need a lemma. Let U\subset \mathbb{R}^n be an open set and f:U\to\mathbb{R} a smooth function. Then there are real numbers \{a_k\} such that f(x_1, \ldots, x_n)-(a_1x_1+a_2x_2+\cdots + a_nx_n) is a Morse function on U. We can also choose \{a_k\} to be arbitrarily small in absolute value.

Let p\in U be a critical point of f. Define h=Jac(f)^T (a smooth map h:U\to\mathbb{R}^n). Then Jac(h)\Big|_p is the Hessian H_f(p). Thus, p is a critical point of h iff det(H_f(p))=0.

By Sard’s Theorem, we can choose a=(a_1, \ldots , a_n)\in\mathbb{R}^n where each a_k have arbitrarily small absolute value such that a is not a critical value of h.

The claim is that \overline{f}(x_1, \ldots , x_n)=f(x_1, \ldots, x_n)-(a_1x_1+\cdots + a_nx_n) is a Morse function on U.

Well, if p is a critical point of \overline{f}, then since \frac{\partial \overline{f}}{\partial x_i}\Big|_p=\frac{\partial f}{\partial x_i}\Big|_p - a_i=0, by the definition of h, we get h(p)=a.

But we chose a to not be a critical value of h. Thus, p is not a critical point of h. So as noted, det(H_f(p))\neq 0. But H_f(p)=H_{\overline{f}}(p), so p is a non-degenerate critical point. Since p was an arbitrary critical point, all critical points are non-degenerate and hence \overline{f} is Morse, completing the proof of the Lemma.

We also need another Lemma. Let K\subset M be a compact subset. Then if g:M\to\mathbb{R} has no degenerate critical points in K, then we can choose \varepsilon >0 small enough so that any C^2 approximation of g also has no degenerate critical points in K.

Since our manifold is closed, it is compact. So we can choose a finite subcover of coordinate charts, and compactly refine it (I’ll do this construction if someone asks in the comments), so that \{U_i\}_1^m cover M and there are compact sets K_i\subset U_i such that \cup K_i=M.

But with this, we can look at any of the U_k, and in these coordinates, g has no degenerate critical points in K\cap K_k (alright, that was probably a poor choice of notation) iff \displaystyle\Big|\frac{\partial g}{\partial x_1}\Big|+\cdots + \Big|\frac{\partial g}{\partial x_n}\Big|+\Big| det(H_g)\Big|>0 for every point in K\cap K_k.

But for a small enough \varepsilon we can definitely still make that inequality hold for any C^2 approximation. Thus we have proved the lemma.

Now let’s do the actual existence proof. Take the U_i, K_i as before. We will inductively build our C^2 approximations on C_l=K_1\cup \cdots \cup K_l. Our base step is to build f_0 on C_0=\emptyset, so we’re done.

For our inductive hypothesis, suppose we have f_{l-1}:M\to\mathbb{R} having no degenerate critical points in C_{l-1}.

Let’s work with the coordinate neighborhood U_l with coordinates (x_i). By the first lemma, there are arbitrarily small numbers \{a_i\} so that f_{l-1}(x_1, \ldots , x_n)-(a_1x_1+\cdots + a_nx_n) is Morse on U_l. But note, we only have a definition on U_l and we need one everywhere.

Let \psi be a bump function that is 1 on K_l and supported in V, where K_l\subset V\subset U_l.

Define f_l=\begin{cases} f_{l-1}-\psi\cdot (a_1x_1+\cdots a_nx_m) & in \ U_l \\  f_{l-1} & outside \ V\end{cases}.

(So I have this same cases problem again, just ignore the “line break” symbol, it is actually readable this time).

This gives us a nice well-defined function on all of M (just need to check the overlaps). Also f_l is our first lemma function on K_l, so it is Morse on K_l and hence has no degenerate critical points there.

Since 0\leq \psi \leq 1 (and we’re on a compact set), we can make \{a_i\} small enough so that f_l is an arbitrarily close C^2 approximation of f_{l-1} (I won’t do this since it is fairly long and tedious, but quite straightforward for the reasons I gave).

But now by the second lemma, since f_{l-1} has no degenerate critical points in C_{l-1}, we have that f_l has no degenerate critical points in C_{l-1} either. We already checked on K_l, and thus there are no deg. critical points on C_{l-1}\cup K_l=C_l.

Thus inductively we can get a Morse function on all of M that is C^2-close to our original smooth function.


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