10-22 Notes
Morse theory allows us to determine topological and geometric properties of manifolds by studying the critical points of special real-valued functions known as Morse functions.Marston Morsed used these functions to understand the homology of a manifold. Stephen Smale and Raoul Bott used Morse theory to study the diffeomorphisms of a manifold and its homotopy type. In this lecture, we list some basic notions of Morse theory and give a proof of the stable/unstable manifold theorem.\begin{defn} Let $M$ be a smooth finite-dimensional compact manifold and let $ f: M \rightarrow \R $ be a smooth function.A critical point of $f$ is a point $ p \in M $ such that the derivative of $f$ at $p$ is zero.The Hessian $\mathcal{H}_f(p)$ of $f$ at $p$ is the symmetric bilinear map
\begin{eqnarray*}
\mathcal{H}_f(p): T_pM \times T_pM \rightarrow \R \\
\mathcal{H}_f(p)(v_1,v_2)=(X_1\cdot(X_2 \cdot f))(p)
\end{eqnarray*}where $X_1,X_2$ are local vector field extensions of $v_1,v_2 \in T_pM$. Given a coordinate system $(x_1,x_2,\ldots,x_m)$ defined on a neighborhood of a critical point $p$ of $f$,the Hessian $\mathcal{H}_f(p)$ is represented by the matrix \begin{eqnarray*}
\left[
\begin{array}{ccccc}
\frac{\partial^2f}{\partial x^{2}_{1}}(p) & &\ldots
& &
\frac{\partial^2f}{\partial x_1 \partial x_m}(p)\\
& \ddots & &
&
\\
\vdots & & \frac{\partial^2f}{\partial x_i\partial
x_j}(p)& & \vdots
\\
& & &
\ddots &
\\
\frac{\partial^2f}{\partial x_m \partial x_1}(p) & &\ldots
& &
\frac{\partial^2f}{\partial x^{2}_{m}}(p)
\end{array}
\right]
\end{eqnarray*}A critical point $p$ is said to be non-degenerate if, given any coordinate system defined near $p$, the determinant of the corresponding Hessian matrix is nonzero. Otherwise, $p$ is said to be degenerate. If the critical point $p$ is nondegenerate, its index is defined to be the number of negative eigenvalues of the Hessian matrix.The function $f $ is called a Morse function if all its critical points are non-degenerate. \end{defn}Every manifold has a Morse function: any smooth real-valued function on a compact manifold $M$ can be smoothly perturbed into a Morse function.Furthermore,it can be shown that the critical points of a Morse function $f$ are isolated. Hence, the set $Crit(f)$ of critical points is finite.
Let $g$ be a Riemannian metric on the compact manifold $M$. Given a Morse function $f$, we consider the flow $\phi_t$ generated by $-\nabla_{g}f$, the negative gradient vector field of $f$. For every critical point $p$ of $f$, we define\begin{eqnarray*} W^{+}_{p} &=& \{x\in M\,\,| \lim_{t \rightarrow \infty} \phi_s(x) = p \} \\ W^{-}_{p} &=& \{x\in M\,\,| \lim_{t \rightarrow -\infty} \phi_s(x) = p \} \end{eqnarray*} These sets are respectively called the stable manifold and unstable manifold of $f$ at $p$.\begin{thm}[Hartman-Grobman(1959),Smale(1962)] $W^{\pm}_{p}$ is a smooth manifold which is diffeomorphic to $\R^{\mu_{\pm}}$ where $\mu_{\pm}$ is the number of positive (negative) eigenvalues of the Hessian of $f$ at $p$. \end{thm}(Perron's method) Locally, $ W^{+}_{p}$ is a set of initial conditions for the differential equation $\dot{x}=-\nabla_{g}f(x)$ to have a "small solution" $ \forall t > 0$.Hence,we will consider all maps $\gamma: [0, \infty) \rightarrow M $ such that,$ \forall t \in [0, \infty)$,$\gamma(t)$ is in a neighborhood $U$ of $p$ and $\lim_{t \rightarrow \infty} \gamma(t)= p $. Then $W^{+ loc}_{p} = \{ \gamma(0) | \gamma \,\, \text{as above} \}$.Since we are working locally , we will consider maps $\gamma:[0,\infty) \rightarrow \R^{n}$ where $0 \in \R^{n} $ corresponds to the critical point $p$.Let $I= [0,\infty)$.Consider the Sobolev spaces $H_{1}= L^{2}_{1}(I,\R^{n})$ and $H_{0}= L^{2}_{1}(I,\R^{n})$ with norms $\|\ \gamma \,\|_{1}= (\int_{0}^{\infty} |\dot{\gamma}|^{2}+|\gamma|^{2})^{1/2}$ and $\|\ \gamma \,\|_{0}= (\int_{0}^{\infty}|\gamma|^{2})^{1/2}$ respectively.\begin{prop} $H_{1}= C^{1/2}(I,\R^{n})$ and $ \forall \gamma \in t $, $\lim_{t \rightarrow \infty} \gamma(t)= 0 $. \end{prop}Proof:Given $\gamma \in H_{1}$ and $t_{2} \,> t_{1} \, >0$, we have\begin{eqnarray*} \gamma(t_{2})-\gamma(t_{1}) = \int_{t_{1}}^{t_{2}} \dot{\gamma}(s)ds \end{eqnarray*}Hence \begin{eqnarray*} |\gamma(t_{2})-\gamma(t_{1})| \le \int_{t_{1}}^{t_{2}} |\dot{\gamma}(s)|ds \le (\int_{t_{1}}^{t_{2}} \dot{\gamma}(s)ds)^{1/2}(t_{2}-t_{1})^{1/2} \le \|\ \gamma \,\|_{1}(t_{2}-t_{1})^{1/2} \end{eqnarray*}It is clear that $\lim_{t \rightarrow \infty} \gamma(t)= 0 $. $\Box$
Consider the function\begin{eqnarray*} F: H_{1} \rightarrow H_{0} \gamma \rightarrow \dot{\gamma}+\nabla_{g}f(\gamma) \end{eqnarray*}$F^{-1}(0)$ is the set of flow lines of $-\nabla_{g}f$.It can be shown that $|\nabla_{g}f(x)| < O(|x|)$ in a sufficiently small open ball $B$ about $0 \in \R^n$.Then on the open subset $\{ \gamma \in H_{1} | \gamma(t) \in B \forall B \}$, we will have\begin{eqnarray*} \int_{0}^{\infty} |\nabla_{g}f(\gamma)|^{2} < C \int_{0}^{\infty}|\gamma|^{2} \end{eqnarray*}$F:U \rightarrow H_{0}$ is continuous (after shrinking $U$ if necessary).Indeed\begin{eqnarray*} \|\ F(\gamma_{2}) -F(\gamma_{2}) \,\|_{0}^{2} \le C_{1} (\int_{0}^{\infty}|\dot{\gamma}_{2}-\dot{\gamma}_{1}|^{2} + \int_{0}^{\infty} |\nabla_{g}f(\gamma_{2})-\nabla_{g}f(\gamma_{1})|^{2}) \le C_{2} (\int_{0}^{\infty} |\dot{\gamma}_{2}-\dot{\gamma}_{1}|^{2}+|\gamma_{2}-\gamma_{1}|^{2}) \end{eqnarray*} It can be shown that $F \in C^{k} \, \, \forall k$ after shrinking the domain if necessary.
