Pseudo-Gradients

Abstract

We define, construct and study pseudo-gradient fields, whose trajectories connect the critical points of a Morse function. These vector fields allow us to define the stable and unstable manifolds of the critical points, which will play an important role. We call attention to the “male property” because of which, for example, there are only finitely many trajectories connecting two critical points with consecutive indices and we prove the existence of pseudo-gradient fields satisfying this property.

Exercises

Exercise 10

Show that the vector fields whose flows are drawn in Figure 2.30 are not pseudo-gradient fields.

Fig. 2.30

Exercise 11

Let V be a manifold of dimension 2 endowed with a Morse function with a unique critical point of index 1. Show that every pseudo-gradient field adapted to this function satisfies the Smale condition.

Exercise 12

We fix an integer m≥2. Find all critical points of the function $f:\mathbf{P}^{1}(\mathbf{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ defined by

$$f([z_0,z_1])=\frac{\left \vert z_0^m+z_1^m\right \vert ^2}{(\left \vert z_0\right \vert ^2+ \left \vert z_1\right \vert ^2)^{m}}=\frac{\left \vert z^m+1\right \vert ^2}{(\left \vert z\right \vert ^2+1)^{m}}$$

(in homogeneous coordinates or in the affine chart z ₁≠0). Verify that for m=2, the function f is not a Morse function.³

We suppose that m≥3. Show that f is a Morse function and has two local maxima: the points 0 and ∞; m local minima: the m-th roots of −1; and m critical points of index 1: the m-th roots of 1.

Hint: We can determine the critical points using the derivatives with respect to z and $\overline {z}$, and then use a second-order Taylor expansion of f(u) with respect to u in the neighborhood of 0 (to study the critical points at 0 and ∞) or the analogous expansion of f(ζ(1+u)) (to study the critical points at ζ with ζ ^m=±1).

Show that there exists a pseudo-gradient field such as that shown (in an affine chart) in Figure 2.31 (for m=3). More generally, see the article [9] in which an analogous function (defined on P ⁿ(C)) plays an important role.

Fig. 2.31