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

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Proper Morse slabs prevent finite-time escape of connecting trajectories

Statement

Let $γ : I \to M$ be a maximal negative-gradient trajectory. If its image is contained in a compact Morse slab $f^{- 1} ([a, b])$ , then $I = R$ . Consequently, this applies to any trajectory already known to have endpoint levels in $[a, b]$ and to remain in that slab; it is not a blanket noncompact-completeness assertion.

Facts & Assumptions

Given: A maximal negative-gradient trajectory $γ : I \to M$ with image in the compact slab $K = f^{- 1} ([a, b])$ .

[F1]

A compact Morse slab is a compact inverse image $f^{- 1} ([a, b])$ (Proper smooth functions and compact Morse slabs).

[F2]

The energy identity makes $f \circ γ$ nonincreasing (A negative-gradient trajectory satisfies the energy identity).

[F3]

Near every point of $M$ , the vector field has unique integral curves on a uniform local time interval (Local existence, uniqueness, and smooth dependence for manifold integral curves).

[F4]

A maximal integral curve cannot have a genuine extension (Through each point there is a unique maximal integral curve).

Proof

technique · direct

1.1

Suppose the right endpoint $T$ of $I$ were finite. The local flow neighbourhoods supplied by [F3] cover the compact set $K$ in [F1], so finitely many suffice; their time radii have a positive minimum $ε$ .

F1F3assume-contra

2.1

Choose $t \in I$ with $T - t < ε$ . Since $γ (t) \in K$ , the corresponding local solution from [F3] extends $γ$ beyond $T$ ; uniqueness identifies it with $γ$ on the overlap, contradicting [F4].

F3F4step 1.1choosedischarge-contradiction

3.1

The same argument at the left endpoint proves $I = R$ . If endpoint levels lie in $[a, b]$ , [F2] verifies the usual monotone trapping in that slab once the trajectory is known to remain between those levels.

F2step 2.1∎

Depends on

Used by

Completeness of a gradient flow is an extra hypothesis on a noncompact manifold Remark

Dependency tree · two levels

14 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

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, §13.1 (standard reference, not scraped)