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Continuation solutions have critical limits and exponential decay

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(f_{s}, g_{s})$ be a continuation datum from $(f^{-}, g^{-})$ to $(f^{+}, g^{+})$ on a closed manifold $M$ and let $u : R \to M$ be a solution of the continuation equation (A regular continuation datum between Morse--Smale pairs). Then the limits $\lim_{s \to - \infty} u (s) = p \in Crit (f^{-}), \lim_{s \to + \infty} u (s) = q \in Crit (f^{+})$ exist. Moreover, in the Morse coordinates of Morse lemma at $p$ and $q$ the curve $u$ converges to $p$ (resp. $q$ ) exponentially fast as $s \to - \infty$ (resp. $s \to + \infty$ ), and $∣ \partial_{s} u (s) ∣_{g_{s}}$ decays exponentially; in particular the energy integral $\int_{R} ∣ \partial_{s} u ∣^{2} d s$ is finite.

Facts & Assumptions

Given: ${A C}_{ω}$ , a closed manifold $M$ , Morse--Smale pairs $(f^{\pm}, g^{\pm})$ in the metric sense, a continuation datum $(f_{s}, g_{s})$ with threshold $S > 0$ , and a solution $u$ of the continuation equation.

[F1]

On a closed manifold, under ${A C}_{ω}$ , every smooth vector field is complete, so the autonomous negative-gradient fields $- \nabla^{g^{\pm}} f^{\pm}$ have global flows; uniqueness of solutions of the autonomous equation identifies reparametrized solution curves with trajectories of that flow (Every smooth vector field on a compact manifold is complete, The fundamental theorem on flows).

[F2]

Under ${A C}_{ω}$ , a negative-gradient trajectory of a Morse function on a compact manifold has a unique $α$ -limit and a unique $ω$ -limit, both critical points (A negative-gradient trajectory on a compact Morse manifold has single critical alpha and omega limits, A Morse trajectory from one critical point to another).

[F3]

The datum is constant on the two half-lines: $(f_{s}, g_{s}) = (f^{-}, g^{-})$ for $s \leq - S$ and $(f_{s}, g_{s}) = (f^{+}, g^{+})$ for $s \geq S$ (A regular continuation datum between Morse--Smale pairs).

[F4]

At a Morse critical point of the actual metric negative gradient, the local stable and unstable disks are tangent to the positive and negative Hessian eigenspaces, and the weighted-path parametrization of those disks gives exponential convergence of the orbits: a trajectory of $- \nabla^{g^{\pm}} f^{\pm}$ that converges to a critical point in forward (resp. backward) time decays exponentially in that time (Stable and unstable manifolds for a Morse gradient with an arbitrary smooth metric, Morse lemma).

Proof

technique · direct

1.1F1F3givenconstruct

On the negative half-line $s \leq - S$ , the continuation equation reads $\partial_{s} u = - \nabla^{g^{-}} f^{-} (u (s))$ by [F3], so there $u$ is a reparametrized solution curve of the autonomous negative-gradient field of $f^{-}$ . By [F1] that field is complete; let $Φ^{-}$ be its global flow and define $γ (t) = Φ_{t}^{-} (u (- S))$ for $t \in R$ . Then $γ$ is a full negative-gradient trajectory of $f^{-}$ , and uniqueness of solutions of the autonomous equation gives $u (s) = γ (s + S)$ for every $s \leq - S$ .

2.1F2step 1.1

By [F2] the trajectory $γ$ has a unique $α$ -limit $p \in Crit (f^{-})$ ; reparametrizing back gives $\lim_{s \to - \infty} u (s) = \lim_{t \to - \infty} γ (t) = p$ .

2.2F1F2F3step 1.1

The same construction on the positive half-line exhibits the restriction of $u$ to $[S, \infty)$ as the terminal piece of a full trajectory of the complete field $- \nabla^{g^{+}} f^{+}$ , whose unique $ω$ -limit is a critical point $q \in Crit (f^{+})$ by [F2]; hence $\lim_{s \to + \infty} u (s) = q$ .

3.1F4step 2.1step 2.2

Let $p$ be the limit of step 2.1. Since $u (s) \to p$ , the curve enters and stays in a sufficiently small Morse chart at $p$ as $s \to - \infty$ , so it is a trajectory of the metric negative gradient converging to $p$ in backward time. By [F4] it lies on the local unstable disk, whose weighted-path parametrization bounds $∥ u (s) - p ∥ \leq C e^{λ s}$ for $s \leq s_{0}$ with constants $C, λ > 0$ in the Morse coordinates of Morse lemma; replacing $s$ by $- s$ and $p$ by $q$ gives the analogous bound $∥ u (s) - q ∥ \leq C^{'} e^{- λ^{'} s}$ for $s \geq s_{1}$ .

4.1step 3.1algebra∎

On the two half-lines $\nabla^{g_{s}} f_{s} = \nabla^{g^{\pm}} f^{\pm}$ is smooth with $\nabla^{g^{\pm}} f^{\pm} (p) = 0$ , hence Lipschitz on the small charts, so $∣ \partial_{s} u ∣_{g_{s}} = ∣ \nabla^{g_{s}} f_{s} (u (s)) ∣_{g_{s}}$ is bounded by a constant times $∥ u (s) - p ∥$ near $p$ and by a constant times $∥ u (s) - q ∥$ near $q$ ; by step 3.1 it decays exponentially on both ends and is bounded on the compact window $[- S, S]$ . Therefore $\int_{R} ∣ \partial_{s} u ∣_{g_{s}}^{2} d s$ is finite.

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Sources

Alexander F. Ritter, Part III Morse Homology (Cambridge lecture notes, complete author PDF, 115 pp.) (standard reference, not scraped)
Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (complete author PDF, 291 pp.) (standard reference, not scraped)
Michele Audin and Mihai Damian, Morse Theory and Floer Homology (complete author PDF of the English book, 628 pp.) (standard reference, not scraped)