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.

Definition 1. Let \(M\) be a smooth finite-dimensional compact manifold and let \(f: M \rightarrow \mathbb{R}\) 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 \[\begin{aligned} \mathcal{H}_f(p): T_pM \times T_pM \rightarrow \mathbb{R}\\ \mathcal{H}_f(p)(v_1,v_2)=(X_1\cdot(X_2 \cdot f))(p) \end{aligned}\]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 \[\begin{aligned} \left[ \begin{array}{ccccc} \frac{\partial^2f}{\partial x^{2}_{1}}(p) & \ldots & & \frac{\partial^2f}{\partial x_1 \partial x_m}(p)\\ & \ddots & & \\ \vdots & \frac{\partial^2f}{\partial x_i\partial x_j}(p)& & \vdots \\ & & \ddots & \\ \frac{\partial^2f}{\partial x_m \partial x_1}(p) & \ldots & & \frac{\partial^2f}{\partial x^{2}_{m}}(p) \end{array} \right] \end{aligned}\]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.

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\[\begin{aligned} W^{+}_{p} &= \{x\in M\,\,| \lim_{t \rightarrow \infty} \phi_s(x) = p \} \\ W^{-}_{p} &= \{x\in M\,\,| \lim_{t \rightarrow -\infty} \phi_s(x) = p \} \end{aligned}\] These sets are respectively called the stable manifold and unstable manifold of \(f\) at \(p\).

Theorem 1 (Hartman-Grobman(1959),Smale(1962)). \(W^{\pm}_{p}\) is a smooth manifold which is diffeomorphic to \(\mathbb{R}^{\mu_{\pm}}\) where \(\mu_{\pm}\) is the number of positive (negative) eigenvalues of the Hessian of \(f\) at \(p\).

(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 \(\gamma:[0,\infty) \rightarrow \mathbb{R}^{n}\) where \(0 \in \mathbb{R}^{n}\) corresponds to the critical point \(p\).Let \(I= [0,\infty)\).Consider the Sobolev spaces \(H_{1}= L^{2}_{1}(I,\mathbb{R}^{n})\) and \(H_{0}= L^{2}_{1}(I,\mathbb{R}^{n})\) 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.

Proposition 1. \(H_{1}= C^{1/2}(I,\mathbb{R}^{n})\) and \(\forall \gamma \in t\), \(\lim_{t \rightarrow \infty} \gamma(t)= 0\).

Proof:Given \(\gamma \in H_{1}\) and \(t_{2} \,> t_{1} \, >0\), we have\[\begin{aligned} \gamma(t_{2})-\gamma(t_{1}) = \int_{t_{1}}^{t_{2}} \dot{\gamma}(s)ds \end{aligned}\]Hence \[\begin{aligned} |\gamma(t_{2})-\gamma(t_{1})| \le \int_{t_{1}}^{t_{2}} |\dot{\gamma}(s)|ds \le (\int_{t_{1}}^{t_{2}} \dot{\gamma}(s)ds)^{1/2}(t_{2}-t_{1})^{1/2} \le \|\ \gamma \,\|_{1}(t_{2}-t_{1})^{1/2} \end{aligned}\]It is clear that \(\lim_{t \rightarrow \infty} \gamma(t)= 0\). \(\Box\)

Consider the function\[\begin{aligned} F: H_{1} \rightarrow H_{0} \gamma \rightarrow \dot{\gamma}+\nabla_{g}f(\gamma) \end{aligned}\]\(F^{-1}(0)\) 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 \(0 \in \mathbb{R}^n\).Then on the open subset \(\{ \gamma \in H_{1} | \gamma(t) \in B \forall B \}\), we will have\[\begin{aligned} \int_{0}^{\infty} |\nabla_{g}f(\gamma)|^{2} < C \int_{0}^{\infty}|\gamma|^{2} \end{aligned}\]\(F:U \rightarrow H_{0}\) is continuous (after shrinking \(U\) if necessary).Indeed\[\begin{aligned} \|\ F(\gamma_{2}) -F(\gamma_{2}) \,\|_{0}^{2} \le C_{1} (\int_{0}^{\infty}|\dot{\gamma}_{2}-\dot{\gamma}_{1}|^{2} + \int_{0}^{\infty} |\nabla_{g}f(\gamma_{2})-\nabla_{g}f(\gamma_{1})|^{2}) \le C_{2} (\int_{0}^{\infty} |\dot{\gamma}_{2}-\dot{\gamma}_{1}|^{2}+|\gamma_{2}-\gamma_{1}|^{2}) \end{aligned}\] It can be shown that \(F \in C^{k} \, \, \forall k\) after shrinking the domain if necessary.