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Finite tail length and point convergence on a compact analytic manifold

Statement

Assume countable choice. On every nonempty connected compact real analytic manifold without boundary with any smooth Riemannian metric, every downward gradient trajectory of a real analytic potential has finite length and converges to one critical point. Analytic manifold, metric, potential and gradient have the conventions in Transfer of the analytic gradient inequality to a smooth metric; no isolation of critical points is assumed.

Facts & Assumptions

[F1]

The compact critical-cluster lemma gives $p$ , $U (p) = L$ , and $q (t_{j}) \to p$ . The transfer and trapping lemmas give a whole tail in a neighborhood satisfying the gradient inequality. The local length estimate bounds length after any later $s$ by $e (s)^{1 - ρ} / [c (1 - ρ)]$ , where $e (s) = U (q (s)) - L \to 0$ . Distance is at most joining-curve length.

[F2]

On a connected Riemannian manifold, $d_{g}$ is a finite metric; in particular it satisfies the triangle inequality (Riemannian distance is a metric).

[F9]

Exact suppliers: Transfer of the analytic gradient inequality to a smooth metric, A critical cluster point for a compact gradient trajectory, The local gradient inequality bounds trajectory length, Trapping a gradient tail near a critical cluster point, The Axiom of Countable Choice (AC_ω), Riemannian distance on a connected manifold, Riemannian distance is a metric, The riemannian distance topology is the manifold topology, Riemannian speed and length.

Proof

Given: The data and hypotheses of the Statement.

Goal: Establish its conclusion.

1.1F1F9given

Apply the first four lemmas to obtain the critical cluster point $p$ , a trapped tail starting at $a$ , and constants $c > 0$ , $0 < ρ < 1$ . For every $s \geq a$ and $b > s$ the length estimate gives $\int_{s}^{b} ∥ q^{'} ∥_{g} d t \leq e (s)^{1 - ρ} / [c (1 - ρ)]$ . Taking the supremum over $b$ proves finite tail length and shows that the bound tends to zero as $s \to \infty$ . The initial segment has finite length because its speed is continuous on the compact interval $[0, a]$ .

2.1F1F2F9step 1.1∎

Fix $s \geq a$ . For cluster times $t_{j} \geq s$ , distance is at most the length of $q ∣_{[s, t_{j}]}$ , hence $d_{g} (q (s), q (t_{j})) \leq e (s)^{1 - ρ} / [c (1 - ρ)]$ . Letting $j \to \infty$ and using $q (t_{j}) \to p$ gives $d_{g} (q (s), p) \leq e (s)^{1 - ρ} / [c (1 - ρ)] \to 0$ . The limit $p$ is critical by the cluster lemma. This proves convergence to a single point without assuming that it is isolated or a local minimum.

Depends on

Used by · 1

Single-point convergence for analytic gradient flows

Dependency tree · 2 levels

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