LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-05

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Away from fixed critical neighborhoods, sufficiently small $C^{1}$ perturbations create no new critical points on a compact manifold

Statement

Let $M$ be a compact smooth manifold, let $f : M \to R$ be Morse, and let $U_{1}, \dots, U_{r}$ be pairwise disjoint critical neighbourhoods as in On a compact manifold, a Morse function has finitely many critical points and a uniform Hessian gap on disjoint critical neighborhoods. Then there is $δ > 0$ such that every smooth function $g$ with $∥ d g - d f ∥_{C^{0}} < δ$ on $M ∖ ⋃_{i} U_{i}$ has no critical points in $M ∖ ⋃_{i} U_{i}$ .

Facts & Assumptions

Given: A compact smooth manifold $M$ , a Morse function $f : M \to R$ , and pairwise disjoint critical neighbourhoods $U_{1}, \dots, U_{r}$ for the critical points of $f$ .

[F1]

A critical point of a smooth function is precisely a zero of its differential (Critical points and critical values of a smooth function).

[L1]

The neighbourhoods $U_{i}$ isolate all critical points of $f$ (On a compact manifold, a Morse function has finitely many critical points and a uniform Hessian gap on disjoint critical neighborhoods).

[A1]

On a compact set, the norm of a continuous cotangent vector field attains its minimum.

Proof

technique · direct

1.1

By [L1], the compact set $K : = M ∖ ⋃_{i} U_{i}$ contains no critical point of $f$ . Hence [F1] gives $d f_{x} \neq 0$ for every $x \in K$ .

F1L1given

2.1

By [A1], the continuous function $x \mapsto ∥ d f_{x} ∥$ attains a positive minimum $m$ on $K$ . Put $δ : = m / 2$ .

A1step 1.1choose

3.1

If $g$ satisfies $∥ d g - d f ∥_{C^{0}} < δ$ on $K$ , then for every $x \in K$ one has $∥ d g_{x} ∥ \geq ∥ d f_{x} ∥ - ∥ d g_{x} - d f_{x} ∥ > m - δ = δ > 0$ . Therefore $d g_{x} \neq 0$ on $K$ , so [F1] shows that $g$ has no critical point in $K$ .

F1step 2.1algebra

4.1

Since $K = M ∖ ⋃_{i} U_{i}$ , this says that sufficiently small $C^{1}$ perturbations create no new critical points outside the chosen critical neighbourhoods.

step 3.1∎

Depends on

Used by

Dependency tree · two levels

8 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)

Away from fixed critical neighborhoods, sufficiently small C1C1 perturbations create no new critical points on a compact manifold