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On a compact manifold, a Morse function has finitely many critical points and a uniform Hessian gap on disjoint critical neighborhoods

Statement

Let $M$ be a compact smooth manifold and let $f : M \to R$ be Morse. Then $f$ has finitely many critical points $p_{1}, \dots, p_{r}$ , and one can choose pairwise disjoint coordinate neighbourhoods $U_{i}$ of $p_{i}$ and a constant $η > 0$ such that:

in the chosen chart on $U_{i}$ , every Hessian matrix $H_{i} (x)$ of $f$ on $\overline{U_{i}}$ satisfies $∥ H_{i} (x) v ∥ \geq 2 η ∥ v ∥$ for every coordinate vector $v$ ; and
every smooth function $g$ that is sufficiently $C^{2}$ -close to $f$ on $\overline{U_{i}}$ has exactly one critical point in $U_{i}$ , and that critical point is nondegenerate.

Facts & Assumptions

Given: A compact smooth manifold $M$ and a Morse function $f : M \to R$ .

[F1]

Every critical point of a Morse function is nondegenerate, and a smooth function is Morse exactly when all of its critical points are nondegenerate (Morse functions and excellent Morse functions, A smooth function is Morse if and only if its differential section is transverse to the zero section).

[L1]

A Morse function on a compact manifold has finitely many critical points (A Morse function on a compact manifold has finitely many critical points).

[L2]

A $C^{1}$ map between Euclidean open sets with invertible derivative at a point is a local diffeomorphism near that point (The Euclidean inverse function theorem).

Proof

technique · direct

1.1

By [L1], the critical set of $f$ is finite; write it as ${p_{1}, \dots, p_{r}}$ . Choose pairwise disjoint coordinate charts $φ_{i} : U_{i}^{'} \to B_{i} \subseteq R^{n_{i}}$ with $φ_{i} (p_{i}) = 0$ , and then shrink to relatively compact subcharts $U_{i} ⋐ U_{i}^{'}$ that are still pairwise disjoint.

L1givenchoose

2.1

If $r = 0$ , take $η : = 1$ ; there are no neighbourhood conditions to check, so the conclusion is vacuous. If instead $\dim M = 0$ , then $M$ is discrete and each $U_{i} : = {p_{i}}$ is a coordinate neighbourhood. Again take $η : = 1$ . The displayed matrix inequality is vacuous on the zero vector space, every point is critical and nondegenerate, and every function on $U_{i}$ has exactly the one critical point $p_{i}$ . Thus the conclusion holds in either boundary case. Henceforth assume $r \geq 1$ and $\dim M \geq 1$ .

step 1.1cases

2.2

Write $F_{i} : = \nabla (f \circ φ_{i}^{- 1})$ on $B_{i}$ . Since $p_{i}$ is nondegenerate by [F1], the derivative $D F_{i} (0)$ is the Hessian matrix at $p_{i}$ and is invertible. By [L2], after shrinking $U_{i}$ again we may assume $F_{i}$ is a diffeomorphism from $φ_{i} (U_{i})$ onto an open neighbourhood of $0$ .

F1L2step 1.1

3.1

Continuity of the Hessian matrices on the compact closures $\overline{U_{i}}$ lets us shrink the $U_{i}$ so that every Hessian matrix $H_{i} (x)$ of $f$ on $\overline{U_{i}}$ stays within half the least singular value of $D F_{i} (0)$ . Hence there is $η_{i} > 0$ such that $∥ H_{i} (x) v ∥ \geq 2 η_{i} ∥ v ∥$ for all $x \in \overline{U_{i}}$ and all coordinate vectors $v$ . Let $η : = \min_{i} η_{i} > 0$ .

step 2.1step 2.2choosealgebra

4.1

If $g$ is sufficiently $C^{2}$ -close to $f$ on $\overline{U_{i}}$ , then the gradient map $\nabla (g \circ φ_{i}^{- 1})$ is $C^{1}$ -close to $F_{i}$ . Because $F_{i}$ is a local diffeomorphism carrying $0$ to $0$ and the Hessian gap from step 3.1 keeps the derivative uniformly invertible on $\overline{U_{i}}$ , the inverse-function argument persists under sufficiently small $C^{1}$ perturbation: the perturbed gradient has a unique zero in $φ_{i} (U_{i})$ , and at that zero its derivative is still invertible. Thus $g$ has exactly one nondegenerate critical point in $U_{i}$ .

L2step 2.2step 3.1

5.1

Steps 1.1, 3.1, and 4.1 give the claimed finite critical set, disjoint critical neighbourhoods, uniform Hessian gap, and local persistence statement.

step 1.1step 3.1step 4.1∎

Depends on

Used by

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)
Marco Gualtieri, Topology I: Smooth Manifolds, Part 10 (standard reference, not scraped)