Persistence of nondegenerate critical points on a manifold

A nondegenerate critical point x of a real function f on a boundaryless finite-dimensional manifold survives every sufficiently small C² perturbation, together with its Morse index.

The perturbation is measured on a fixed compact coordinate set. Fix a chart e of the C² maximal atlas around x. For every sufficiently small radius r there is δ > 0 such that each function g, C² on the inverse image K of the closed coordinate ball of radius r around the chart image of x, whose coordinate differential at that point has norm at most δ and whose coordinate second derivative is within δ of that of f on the ball, has exactly one critical point in K. That critical point is nondegenerate, has the Morse index of x, and its position and critical value are controlled by the coordinate differential of g at the chart image of x (IsManifoldNondegenerateCriticalPoint.exists_criticalPoint_perturbation_manifoldMorseIndex_eq).

For C^n maps with n ≥ 2 and the weak Whitney topology, generated by compact chart derivative tests, this says that x has arbitrarily small compact neighbourhoods K such that every map g near f has exactly one critical point in K; it is nondegenerate with the Morse index of x, and it tends to x, with critical value tending to f x, as g tends to f (TauCeti.IsManifoldNondegenerateCriticalPoint.exists_mem_nhds_eventually_criticalPoint).

Both statements are read off from the vector-space theorem TauCeti.IsNondegenerateCriticalPoint.exists_criticalPoint_perturbation_morseIndex_eq in one chart, using that nondegeneracy, criticality and the Morse index may be computed in any chart of the maximal atlas.

Nondegeneracy of all critical points in a compact set K is likewise stable: if every critical point of f in K is nondegenerate, then so is every critical point in K of each map near f (TauCeti.eventually_forall_isManifoldNondegenerateCriticalPoint). Near a point of K all critical points of f are nondegenerate, since a regular point has a neighbourhood free of critical points and a nondegenerate critical point is isolated. On a closed coordinate ball around it, the vector-space theorem TauCeti.HasNondegenerateCriticalPointsOn.exists_pos_forall_hasNondegenerateCriticalPointsOn applies to the uniformly C²-close coordinate expressions of nearby maps, and finitely many such balls cover K. Consequently, for n ≥ 2, the C^n functions with only nondegenerate critical points in K form an open set (TauCeti.isOpen_setOf_forall_isManifoldNondegenerateCriticalPoint), and so do the smooth functions which are Morse on K (TauCeti.isOpen_setOf_isMorseOn); on a compact manifold so do the Morse functions (TauCeti.isOpen_setOf_isMorse). This is the openness half of the genericity of Morse functions.

References

• M. Audin and M. Damian, Morse Theory and Floer Homology, Springer Universitext, 2014, Chapter 1.
• M. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976, Chapter 6, §1, for the openness of Morse functions on compact sets.
• J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press, 1965, §2.

theorem TauCeti.IsManifoldNondegenerateCriticalPoint.exists_criticalPoint_perturbation_manifoldMorseIndex_eq {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] {x : M} {f : M → ℝ} {e : OpenPartialHomeomorph M H} (he : e ∈ IsManifold.maximalAtlas I 2 M) (hx : x ∈ e.source) (h : IsManifoldNondegenerateCriticalPoint I f x) : ∃ (K : ℝ), ∀ᶠ (r : ℝ) in nhdsWithin 0 (Set.Ioi 0), Metric.closedBall (↑(e.extend I) x) r ⊆ (e.extend I).target ∧ ∃ δ > 0, ∀ (g : M → ℝ), (∀ y ∈ ↑(e.extend I).symm '' Metric.closedBall (↑(e.extend I) x) r, ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 g y) → ‖fderiv ℝ (g ∘ ↑(e.extend I).symm) (↑(e.extend I) x)‖ ≤ δ → (∀ z ∈ Metric.closedBall (↑(e.extend I) x) r, ‖fderiv ℝ (fderiv ℝ (g ∘ ↑(e.extend I).symm)) z - fderiv ℝ (fderiv ℝ (f ∘ ↑(e.extend I).symm)) z‖ ≤ δ) → ∃ y ∈ ↑(e.extend I).symm '' Metric.closedBall (↑(e.extend I) x) r, IsManifoldNondegenerateCriticalPoint I g y ∧ manifoldMorseIndex I g y = manifoldMorseIndex I f x ∧ dist (↑(e.extend I) y) (↑(e.extend I) x) ≤ K * ‖fderiv ℝ (g ∘ ↑(e.extend I).symm) (↑(e.extend I) x)‖ ∧ |g y - g x| ≤ K * ‖fderiv ℝ (g ∘ ↑(e.extend I).symm) (↑(e.extend I) x)‖ ^ 2 ∧ ∀ z ∈ ↑(e.extend I).symm '' Metric.closedBall (↑(e.extend I) x) r, z ∈ criticalPoints I g → z = y

Persistence of a nondegenerate critical point on a manifold, in a chart.

Let x be a nondegenerate critical point of f on a boundaryless finite-dimensional manifold and e a chart of the C² maximal atlas at x. For every sufficiently small r, the closed ball of radius r around e.extend I x lies in the chart target, and there is δ > 0 with the following property. Let g be C² on the inverse image of that ball, with coordinate differential of norm at most δ at e.extend I x, and with coordinate second derivative within δ of that of f on the ball. Then g has exactly one critical point y in the inverse image of the ball. It is nondegenerate with the Morse index of x, its chart image lies within K * ‖dg‖ of that of x, and |g y - g x| ≤ K * ‖dg‖ ^ 2, where dg is the coordinate differential of g at e.extend I x and K depends only on f, x and e.

theorem TauCeti.IsManifoldNondegenerateCriticalPoint.exists_mem_nhds_eventually_criticalPoint {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] {x : M} {n : WithTop ℕ∞} [IsManifold I n M] {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ n} (hn : 2 ≤ n) (h : IsManifoldNondegenerateCriticalPoint I (⇑f) x) {V : Set M} (hV : V ∈ nhds x) : ∃ K ∈ nhds x, K ⊆ V ∧ IsCompact K ∧ ∀ U ∈ nhds x, ∀ W ∈ nhds (f x), ∀ᶠ (g : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ n) in nhds f, ∃ y ∈ K ∩ U, g y ∈ W ∧ IsManifoldNondegenerateCriticalPoint I (⇑g) y ∧ manifoldMorseIndex I (⇑g) y = manifoldMorseIndex I (⇑f) x ∧ ∀ z ∈ K, z ∈ criticalPoints I ⇑g → z = y

Persistence of a nondegenerate critical point in the weak Whitney topology.

Let x be a nondegenerate critical point of a C^n function f, n ≥ 2, on a boundaryless finite-dimensional manifold. Inside every neighbourhood V of x there is a compact neighbourhood K of x such that every map g near f in the weak Whitney topology has exactly one critical point y in K; it is nondegenerate, has the Morse index of x, and it lies in any prescribed neighbourhood U of x, with g y in any prescribed neighbourhood W of f x, once g is close enough to f.

theorem TauCeti.eventually_forall_isManifoldNondegenerateCriticalPoint {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] {n : WithTop ℕ∞} [IsManifold I n M] {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ n} (hn : 2 ≤ n) {K : Set M} (hK : IsCompact K) (hf : ∀ x ∈ K, x ∈ criticalPoints I ⇑f → IsManifoldNondegenerateCriticalPoint I (⇑f) x) : ∀ᶠ (g : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ n) in nhds f, ∀ x ∈ K, x ∈ criticalPoints I ⇑g → IsManifoldNondegenerateCriticalPoint I (⇑g) x

Nondegeneracy on a compact set persists in the weak Whitney topology.

Let f be a C^n function, n ≥ 2, on a boundaryless finite-dimensional manifold, all of whose critical points in a compact set K are nondegenerate. Then every map near f in the weak Whitney topology again has only nondegenerate critical points in K.

theorem TauCeti.isOpen_setOf_forall_isManifoldNondegenerateCriticalPoint {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] {n : WithTop ℕ∞} [IsManifold I n M] (hn : 2 ≤ n) {K : Set M} (hK : IsCompact K) : IsOpen {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ n | ∀ x ∈ K, x ∈ criticalPoints I ⇑f → IsManifoldNondegenerateCriticalPoint I (⇑f) x}

For a compact set K and n ≥ 2, the C^n functions all of whose critical points in K are nondegenerate form an open set in the weak Whitney topology.

theorem TauCeti.isOpen_setOf_forall_mem_criticalPoints_isManifoldNondegenerateCriticalPoint {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] {n : WithTop ℕ∞} [IsManifold I n M] [CompactSpace M] (hn : 2 ≤ n) : IsOpen {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ n | ∀ x ∈ criticalPoints I ⇑f, IsManifoldNondegenerateCriticalPoint I (⇑f) x}

On a compact manifold and for n ≥ 2, the C^n functions all of whose critical points are nondegenerate form an open set in the weak Whitney topology.

theorem TauCeti.isOpen_setOf_isMorseOn {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] {K : Set M} (hK : IsCompact K) : IsOpen {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑⊤ | IsMorseOn I (⇑f) K}

For a compact set K, the smooth functions which are Morse on K form an open set in the weak Whitney topology.

theorem TauCeti.isOpen_setOf_isMorse {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [CompactSpace M] : IsOpen {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑⊤ | IsMorse I ⇑f}

Morse functions on a compact manifold form an open set in the weak Whitney topology on smooth functions.