LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Every precompact end-limit point of a negative-gradient trajectory is critical

Statement

Every point of a nonempty precompact $α$ - or $ω$ -limit set of a full negative-gradient trajectory is a critical point of $f$ .

Facts & Assumptions

Given: A full negative-gradient trajectory $γ$ with a precompact positive or negative tail.

[F1]

Its tail-limit set is nonempty and flow-invariant (Precompact trajectory tails have nonempty compact connected flow-invariant limit sets).

[F2]

Along $γ$ , $(f \circ γ)^{'} = - ∥ {grad}_{g} f ∥_{g}^{2}$ (A negative-gradient trajectory satisfies the energy identity).

[F3]

A noncritical point has nonzero gradient (The Riemannian gradient vanishes exactly at the critical points).

Proof

technique · direct

1.1F2F1given

On a precompact positive tail, $f \circ γ$ is decreasing by [F2] and bounded below because $f$ is continuous on its compact closure. It therefore has a finite limit $ℓ$ ; every point of $ω (γ)$ is a limit of tail values and has $f = ℓ$ . The same argument, with increasing time reversed, applies to $α (γ)$ .

1.2F2F3assume-contra

Let $z$ lie in either limit set. If $z$ were noncritical, [F3] and [F2] would give a sufficiently short positive orbit segment from $z$ on which $f$ strictly decreases.

2.1F1step 1.1step 1.2discharge-contradiction∎

By [F1] the entire short segment in step 1.2 remains in the same limit set, whereas step 1.1 makes $f$ constant there. This contradiction proves that $z$ is critical.

Depends on

Used by

A negative-gradient trajectory on a compact Morse manifold has single critical alpha and omega limits Lemma

Dependency tree · two levels

9 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

Liviu I. Nicolaescu, An Invitation to Morse Theory, Lemma 2.4.1 (standard reference, not scraped)