Persistence of nondegenerate critical points under small C² perturbations

A nondegenerate critical point x of f survives every sufficiently small C² perturbation: on each sufficiently small closed ball around x there is a threshold δ > 0 such that every function g which is C² on the ball, whose differential at x has norm at most δ, and whose second derivative stays within δ of that of f on the ball, has exactly one critical point in the ball. That critical point is nondegenerate, its distance to x is linearly controlled by ‖fderiv ℝ g x‖, and its critical value differs from g x by a quadratic amount in ‖fderiv ℝ g x‖. In finite dimensions its Morse index is that of x.

Unlike the continuation theorem TauCeti.IsNondegenerateCriticalPoint.exists_contDiffAt_criticalPoint_continuation, no parametrised family is involved: the perturbation is an arbitrary function, and the size of the admissible perturbation is uniform on the chosen ball. A perturbation that is C²-close to f on the ball satisfies the hypotheses, since the differential of f vanishes at x.

The existence and uniqueness of the perturbed critical point follow from Mathlib's quantitative inverse function theorem: on the ball the differential of g approximates the invertible Hessian of f at x, so it is injective there and its image contains an explicit ball around fderiv ℝ g x. The Morse index is preserved because the negative index of a nondegenerate symmetric bilinear form is locally constant.

Nondegeneracy of all critical points on a compact set K persists as well: there is δ > 0 such that every function g, C² on K, whose first and second derivatives stay within δ of those of f on K, again has only nondegenerate critical points on K. At a point of K either the differential of f is far from zero, and then so is that of g, or the second derivative of f is far from the closed set of non-invertible operators, and then that of g is invertible. Compactness makes the two alternatives uniform.

Main declarations

• TauCeti.IsNondegenerateCriticalPoint.exists_criticalPoint_perturbation: in a Banach space, a small perturbation has a unique critical point near x; it is nondegenerate, with controlled location and critical value.
• TauCeti.IsNondegenerateCriticalPoint.exists_pos_forall_morseIndex_eq: in finite dimensions, a C² function whose Hessian at a point is close to the Hessian of f at x has the same Morse index there.
• TauCeti.IsNondegenerateCriticalPoint.exists_criticalPoint_perturbation_morseIndex_eq: in finite dimensions the perturbed critical point moreover has the Morse index of x.
• TauCeti.HasNondegenerateCriticalPointsOn.exists_pos_forall_hasNondegenerateCriticalPointsOn: in a Banach space, a function C²-close to f on a compact set on which f has only nondegenerate critical points has only nondegenerate critical points there.

References

• M. Audin and M. Damian, Morse Theory and Floer Homology, Springer Universitext, 2014, Chapter 1.
• J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press, 1965, §2.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_criticalPoint_perturbation {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} [CompleteSpace E] (h : IsNondegenerateCriticalPoint f x) : ∃ (K : ℝ), ∀ᶠ (r : ℝ) in nhdsWithin 0 (Set.Ioi 0), ∃ δ > 0, ∀ (g : E → ℝ), (∀ y ∈ Metric.closedBall x r, ContDiffAt ℝ 2 g y) → ‖fderiv ℝ g x‖ ≤ δ → (∀ y ∈ Metric.closedBall x r, ‖fderiv ℝ (fderiv ℝ g) y - fderiv ℝ (fderiv ℝ f) y‖ ≤ δ) → ∃ y ∈ Metric.closedBall x r, IsNondegenerateCriticalPoint g y ∧ dist y x ≤ K * ‖fderiv ℝ g x‖ ∧ |g y - g x| ≤ K * ‖fderiv ℝ g x‖ ^ 2 ∧ ∀ z ∈ Metric.closedBall x r, fderiv ℝ g z = 0 → z = y

Persistence of a nondegenerate critical point under small C² perturbations.

Let x be a nondegenerate critical point of f on a Banach space. For every sufficiently small radius r there is δ > 0 with the following property. If g is C² at every point of the closed ball of radius r around x, the differential of g at x has norm at most δ, and the second derivatives of g and f differ by at most δ on that ball, then g has a critical point y in the ball, and it is the only one there. It is nondegenerate, lies within K * ‖fderiv ℝ g x‖ of x, and its critical value differs from g x by at most K * ‖fderiv ℝ g x‖ ^ 2, for a constant K depending only on f and x.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_pos_forall_morseIndex_eq {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} [FiniteDimensional ℝ E] (h : IsNondegenerateCriticalPoint f x) : ∃ ε > 0, ∀ ⦃g : E → ℝ⦄ ⦃y : E⦄, ContDiffAt ℝ 2 g y → ‖fderiv ℝ (fderiv ℝ g) y - fderiv ℝ (fderiv ℝ f) x‖ < ε → morseIndex g y = morseIndex f x

The Morse index is stable under small changes of the Hessian.

In finite dimensions, there is ε > 0 such that every function g which is C² at a point y and whose Hessian at y is within ε of the Hessian of f at the nondegenerate critical point x has the Morse index of f at x. The point y need not be critical.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_criticalPoint_perturbation_morseIndex_eq {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} [FiniteDimensional ℝ E] (h : IsNondegenerateCriticalPoint f x) : ∃ (K : ℝ), ∀ᶠ (r : ℝ) in nhdsWithin 0 (Set.Ioi 0), ∃ δ > 0, ∀ (g : E → ℝ), (∀ y ∈ Metric.closedBall x r, ContDiffAt ℝ 2 g y) → ‖fderiv ℝ g x‖ ≤ δ → (∀ y ∈ Metric.closedBall x r, ‖fderiv ℝ (fderiv ℝ g) y - fderiv ℝ (fderiv ℝ f) y‖ ≤ δ) → ∃ y ∈ Metric.closedBall x r, IsNondegenerateCriticalPoint g y ∧ morseIndex g y = morseIndex f x ∧ dist y x ≤ K * ‖fderiv ℝ g x‖ ∧ |g y - g x| ≤ K * ‖fderiv ℝ g x‖ ^ 2 ∧ ∀ z ∈ Metric.closedBall x r, fderiv ℝ g z = 0 → z = y

Persistence of a nondegenerate critical point with its Morse index.

In finite dimensions, the unique critical point of a small C² perturbation provided by exists_criticalPoint_perturbation moreover has the Morse index of f at x.

theorem TauCeti.HasNondegenerateCriticalPointsOn.exists_pos_forall_hasNondegenerateCriticalPointsOn {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} [CompleteSpace E] {K : Set E} (hK : IsCompact K) (hf₁ : ContinuousOn (fderiv ℝ f) K) (hf₂ : ContinuousOn (fderiv ℝ (fderiv ℝ f)) K) (hM : HasNondegenerateCriticalPointsOn f K) : ∃ δ > 0, ∀ (g : E → ℝ), (∀ x ∈ K, ContDiffAt ℝ 2 g x) → (∀ x ∈ K, ‖fderiv ℝ g x - fderiv ℝ f x‖ ≤ δ) → (∀ x ∈ K, ‖fderiv ℝ (fderiv ℝ g) x - fderiv ℝ (fderiv ℝ f) x‖ ≤ δ) → HasNondegenerateCriticalPointsOn g K

Nondegeneracy on a compact set persists under small C² perturbations.

Let f have only nondegenerate critical points on a compact subset K of a Banach space, with first and second derivatives continuous on K. Then there is δ > 0 such that every function g which is C² at each point of K, and whose first and second derivatives are within δ of those of f at each point of K, again has only nondegenerate critical points on K.