Broken Morse chains and finite index-one trajectory spaces from compact sublevels

Suppose every sublevel of a finite-dimensional Morse function is compact. Every forward negative-gradient orbit stays in the sublevel of its initial point, so it converges to a critical point. A point in the closure of an unstable manifold also has a backward orbit in the sublevel of its originating critical point; this orbit converges to a critical point as well.

Under Morse--Smale transversality, these convergence facts also produce finite chains of nonconstant connecting orbits forced by stable- or unstable-set closure incidence, with the number of components bounded by the Morse-index drop. They do not assert convergence of a sequence of trajectories to a broken chain.

These two convergence facts discharge the global endpoint hypotheses in the compactness theorem for index-one trajectory slices. The slice at an intermediate value is a closed subset of a compact level, and Morse--Smale transversality makes it discrete. It therefore contains only finitely many unparametrized trajectories, the finiteness needed for the Morse differential.

The compact-sublevel hypothesis is the usual properness condition for a bounded-below Morse function on Euclidean space. On a compact manifold, compactness comes from the ambient space instead.

Reference

M. Audin and M. Damian, Morse Theory and Floer Homology, Springer Universitext, 2014, Chapter 3 (compactness of spaces of trajectories).

theorem TauCeti.exists_stableSet_of_isCompact_sublevel {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {K : NNReal} (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hM : HasNondegenerateCriticalPointsOn f Set.univ) (y : E) (hcompact : IsCompact {z : E | f z ≤ f y}) : ∃ (q : E), IsNondegenerateCriticalPoint f q ∧ y ∈ (negativeGradientFlow f hf).stableSet q

Every forward orbit of a globally Morse function converges to a nondegenerate critical point when the sublevel at its initial value is compact.

theorem TauCeti.exists_unstableSet_of_mem_closure_unstableSet_of_isCompact_sublevel {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {K : NNReal} (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hM : HasNondegenerateCriticalPointsOn f Set.univ) {p y : E} (hcompact : IsCompact {z : E | f z ≤ f p}) (hy : y ∈ closure ((negativeGradientFlow f hf).unstableSet p)) : ∃ (a : E), IsNondegenerateCriticalPoint f a ∧ y ∈ (negativeGradientFlow f hf).unstableSet a

Every backward orbit starting in the closure of an unstable manifold converges to some nondegenerate critical point when the sublevel at the originating point is compact.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_isChain_of_mem_closure_stableSet_inter_stableSet_of_isCompact_sublevel {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {K : NNReal} {x q : E} (hx : IsNondegenerateCriticalPoint f x) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hM : HasNondegenerateCriticalPointsOn f Set.univ) (hcompact : ∀ (c : ℝ), IsCompact {z : E | f z ≤ c}) (hqx : q ≠ x) (hy : (closure ((negativeGradientFlow f hf).stableSet q) ∩ (negativeGradientFlow f hf).stableSet x).Nonempty) (htr : IsMorseSmale f hf) : ∃ (l : List E), List.IsChain (fun (a b : E) => a ≠ b ∧ ((negativeGradientFlow f hf).unstableSet a ∩ (negativeGradientFlow f hf).stableSet b).Nonempty) (x :: l) ∧ (x :: l).getLast ⋯ = q ∧ (∀ a ∈ x :: l, IsNondegenerateCriticalPoint f a) ∧ l.length + morseIndex f q ≤ morseIndex f x

For a globally Morse function with compact sublevels and Morse--Smale transversality, a stable-set closure incidence produces a finite chain x :: l from x to q of nonconstant connecting orbits between nondegenerate critical points. Compactness supplies forward convergence, and l.length is at most morseIndex f x - morseIndex f q.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_isChain_of_mem_closure_unstableSet_inter_unstableSet_of_isCompact_sublevel {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {K : NNReal} {x p : E} (hx : IsNondegenerateCriticalPoint f x) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hM : HasNondegenerateCriticalPointsOn f Set.univ) (hcompact : IsCompact {z : E | f z ≤ f p}) (hpx : p ≠ x) (hy : (closure ((negativeGradientFlow f hf).unstableSet p) ∩ (negativeGradientFlow f hf).unstableSet x).Nonempty) (htr : IsMorseSmale f hf) : ∃ (l : List E), List.IsChain (fun (a b : E) => a ≠ b ∧ ((negativeGradientFlow f hf).unstableSet a ∩ (negativeGradientFlow f hf).stableSet b).Nonempty) (p :: l) ∧ (p :: l).getLast ⋯ = x ∧ (∀ a ∈ p :: l, IsNondegenerateCriticalPoint f a) ∧ l.length + morseIndex f x ≤ morseIndex f p

For a globally Morse function with Morse--Smale transversality and compact sublevel at f p, an unstable-set closure incidence produces a finite chain p :: l from p to x of nonconstant connecting orbits between nondegenerate critical points. Compactness supplies backward convergence, and l.length is at most morseIndex f p - morseIndex f x.

theorem TauCeti.finite_unstableSet_inter_stableSet_inter_level_of_isCompact_sublevel {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {K : NNReal} (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hM : HasNondegenerateCriticalPointsOn f Set.univ) (hcompact : ∀ (c : ℝ), IsCompact {z : E | f z ≤ c}) {p q : E} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hind : morseIndex f p = morseIndex f q + 1) {c : ℝ} (hc : f q < c ∧ c < f p) (htr : IsMorseSmale f hf) : ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q ∩ {z : E | f z = c}).Finite

An index-one Morse--Smale trajectory space has finitely many trajectories when the Morse function has compact sublevels. A trajectory is represented by its unique point at the level f = c, where f q < c < f p.