Level slices of transverse Morse trajectories

Let p ≠ q be nondegenerate critical points of a globally C² function f with globally Lipschitz gradient on a finite-dimensional real inner product space. A negative-gradient trajectory from p to q crosses each intermediate level {f = c} exactly once, so the level slice W^u(p) ∩ W^s(q) ∩ {f = c} is in bijection with the space of unparametrized trajectories from p to q (Flow.IsNegativeGradient.bijOn_quotient_mk_unstableSet_inter_stableSet_level). This file shows that, where W^u(p) and W^s(q) meet transversally, the slice is an embedded C¹ submanifold of dimension morseIndex f p - morseIndex f q - 1.

The level {f = c} is cut transversally: the velocity -∇f y of the trajectory through a point y of the slice is tangent to W^u(p) ∩ W^s(q), while df_y(-∇f y) = -‖∇f y‖² ≠ 0. Hence the tangent space of the slice at y is the tangent space of W^u(p) ∩ W^s(q) cut by ker df_y, one dimension less.

When the Morse indices differ by one, the slice is therefore discrete under the Morse–Smale condition: its points are the isolated trajectories that the Morse differential counts. Finiteness of the count needs, in addition, compactness of the slice.

Main results

• TauCeti.span_tangentConeAt_unstableSet_inter_stableSet_sup_ker_fderiv_eq_top: every level of f is transverse to W^u(p) ∩ W^s(q).
• TauCeti.IsNondegenerateCriticalPoint.exists_unstableSet_inter_stableSet_level_chart: under transversality, the level slice is flattened by a C¹ chart with C¹ inverse.
• TauCeti.IsNondegenerateCriticalPoint.span_tangentConeAt_unstableSet_inter_stableSet_level: its tangent space is the tangent space of W^u(p) ∩ W^s(q) cut by ker df_y.
• finrank_span_tangentConeAt_unstableSet_inter_stableSet_level_add_morseIndex (in the namespace TauCeti.IsNondegenerateCriticalPoint): the slice has dimension morseIndex f p - morseIndex f q - 1.
• TauCeti.IsNondegenerateCriticalPoint.isDiscrete_unstableSet_inter_stableSet_level: if the Morse indices differ by one and the Morse–Smale condition holds along the slice, then the slice is discrete.

References

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

theorem TauCeti.span_tangentConeAt_unstableSet_inter_stableSet_sup_ker_fderiv_eq_top {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} (hf : LipschitzWith K (gradient f)) (hpq : p ≠ q) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q) y) ⊔ (↑(fderiv ℝ f y)).ker = ⊤

Levels are transverse to connecting trajectories. At a point y of W^u(p) ∩ W^s(q) with p ≠ q, the tangent space of W^u(p) ∩ W^s(q) and the kernel of df_y, the tangent space of the level of f through y, span the whole space.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_unstableSet_inter_stableSet_level_chart {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} {c : ℝ} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hpq : p ≠ q) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) (htr : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) (hyc : f y = c) : ∃ (e : OpenPartialHomeomorph E E), y ∈ e.source ∧ (∀ z ∈ e.source, ContDiffAt ℝ 1 (↑e) z) ∧ (∀ z ∈ e.target, ContDiffAt ℝ 1 (↑e.symm) z) ∧ ∀ z ∈ e.source, z ∈ (negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q ∩ {x : E | f x = c} ↔ ↑e z ∈ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊓ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) ⊓ (↑(fderiv ℝ f y)).ker

The level slice of a transverse intersection is an embedded submanifold. If the unstable set of a Morse critical point p and the stable set of a different Morse critical point q meet transversally at y, and f y = c, then near y the level slice W^u(p) ∩ W^s(q) ∩ {f = c} is flattened by a C¹ chart with C¹ inverse onto the intersection of the tangent spaces of W^u(p) and W^s(q) with ker df_y.

theorem TauCeti.IsNondegenerateCriticalPoint.span_tangentConeAt_unstableSet_inter_stableSet_level {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} {c : ℝ} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hpq : p ≠ q) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) (htr : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) (hyc : f y = c) : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q ∩ {x : E | f x = c}) y) = Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊓ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) ⊓ (↑(fderiv ℝ f y)).ker

The tangent space of a transverse level slice. If the unstable set of a Morse critical point p and the stable set of a different Morse critical point q meet transversally at y, and f y = c, then the tangent space of the level slice W^u(p) ∩ W^s(q) ∩ {f = c} at y is the intersection of the tangent spaces of W^u(p) and W^s(q) with ker df_y.

theorem TauCeti.IsNondegenerateCriticalPoint.finrank_span_tangentConeAt_unstableSet_inter_stableSet_level_add_morseIndex {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} {c : ℝ} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hpq : p ≠ q) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) (htr : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) (hyc : f y = c) : Module.finrank ℝ ↥(Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q ∩ {x : E | f x = c}) y)) + morseIndex f q + 1 = morseIndex f p

The level slice has dimension the index difference minus one. If the unstable set of a Morse critical point p and the stable set of a different Morse critical point q meet transversally at y, then the tangent space at y of the level slice W^u(p) ∩ W^s(q) ∩ {f = f y} has dimension morseIndex f p - morseIndex f q - 1.

theorem TauCeti.IsNondegenerateCriticalPoint.isDiscrete_unstableSet_inter_stableSet_level {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q : E} {K : NNReal} {c : ℝ} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hind : morseIndex f p = morseIndex f q + 1) (htr : ∀ y ∈ (negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q ∩ {x : E | f x = c}, Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) : IsDiscrete ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q ∩ {x : E | f x = c})

Index-difference-one level slices are discrete. If the Morse indices of p and q differ by one and the unstable set of p meets the stable set of q transversally at every point of the level slice W^u(p) ∩ W^s(q) ∩ {f = c}, then the slice is discrete: each of the trajectories it parametrizes is isolated.