On the Morse Index of Critical Points in the Viscosity Method

We show that in viscous approximations of functionals defined on Finsler manifolds, it is possible to construct suitable sequences of critical points of these approximations satisfying the expected Morse index bounds as in Lazer-Solimini's theory, together with the entropy condition of Michael Struwe.


Introduction
In this paper, we want to show that one can construct critical points of the right index depending on the dimension of the admissible min-max family in the framework of the viscosity method. Namely, we fix a C 2 Finsler manifold X and we consider a C 2 function F : X → R, for which one aims at constructing (unstable) critical points. We further fix some d-dimensional compact manifold M d with boundary ∂M d = B d−1 = ∅, and a continuous map h : B d−1 → X, and we call the subset A ⊂ P(X) a d-dimensional admissible family (relative to (M d  We shall generalise this example later and introduce additional min-max families in Section 2.2. In particular, notice that A is stable under homeomorphisms isotopic to the identity preserving the boundary h(B d−1 ) ⊂ X. Then the min-max level associated to F and A , denoted here by β(F, A ) or (β(A ) when there is no ambiguity in the choice of F ) Assuming that the min-max is non-trivial in the following sense this is a very classical theorem of Palais ([19]) that there exists a critical point x ∈ K(F ) of F such that F (x) = β(A ), provided F satisfies the celebrated Palais-Smale (PS) condition. Now, we assume furthermore that X is a Finsler-Hilbert manifold and that the linear map ∇ 2 F (x) : T x X → T x X is a Fredholm operator at every critical point x ∈ K(F ). We also define the index Ind F (x) ∈ N (resp. the nullity Null F (x)) of a critical point x ∈ K(F ) of F as the number (with multiplicity) of negative eigenvalues (resp. as the multiplicity of the 0-eigenvalue) of the Fredholm operator ∇ 2 F (x) : T x X → T x X.
In this setting, it was subsequently proved by Lazer and Solimini ([12]) that it is possible to find a critical point x * ∈ K(F ) (a priori different from x) such that we get the following index bound (1.1) For different types of min-max family, it is also possible to obtain a one-sided bound or a two-sided estimate In particular, if F is non-degenerate at x, we obtain a critical point for the third kind of families of index exactly equal to d (to be defined in Section 2.2). For min-max families defined with respect to homology classes, the two-sided estimate was first obtained by Claude Viterbo ([35]). Now, in the framework of the viscosity method (see [15] for a general introduction on the subject), the function F does not satisfy the Palais-Smale condition (take for example minimal or Willmore surfaces), and one wishes to construct critical points of F by approaching F by a more coercive function for which we can apply the previous standard methods. We let G : X → R + be a C 2 function and we define for all σ > 0 the C 2 function F σ = F + σ 2 G, and we assume that for all σ > 0, the function F σ : X → R verifies the Palais-Smale condition. Furthermore, we denote for all σ ≥ 0 (so that β(0) = β(A )) In particular, the previous theory applies and we can find for all σ > 0 a critical point x σ of F σ of the right index. Then this is a case-by-case analysis to show that the bounds carry one as σ → 0 (see [25] for minimal surfaces and [14] for Willmore surfaces). However, the first problem which might occur (and actually the only one) is to loose energy in the approximation part, i.e. to have for some sequence {σ k } k∈N ⊂ (0, ∞) converging towards 0 and some sequence {x k } k∈N ⊂ X of critical points associated to {F σ k } k∈N (i.e. such that x k ∈ K(F σ k , β(σ k )) for all k ∈ N) There are some explicit examples of such failure (see e.g. [15] for examples for geodesics and minimal surfaces), but Michael Struwe found that this was possible to overcome this difficulty through what is called Struwe's monotonicity trick (see [32], [33]). In our setting, the corresponding theorem is the following (see [15] or [23] for a proof).
Theorem ( * ). Let (X, · ) be a complete C 1 Finsler manifold. Let F σ : X → R be a family of C 1 functions for all σ ∈ [0, 1] such that for all x ∈ X, is C 1 and increasing. Assume furthermore that there exists C ∈ L ∞ loc ((0, 1)), δ ∈ L ∞ loc (R + ) going to 0 at 0, and f ∈ L ∞ loc (R) such that for all 0 < σ, τ < 1 and for all x ∈ X, Finally, assume that for σ > 0 the function F σ satisfies the Palais-Smale condition. Let A be an admissible family of min-max of X and denote Then there exists a sequence {σ k } k∈N ⊂ (0, ∞) and {x k } k∈N ⊂ X such that Furthermore, for all k ∈ N, the critical point x k satisfies the following entropy condition Now, one would like to merge the index bound of Lazer and Solimini with Struwe's monotonicity trick, which requires a new argument (we refer to Section 2.2 for the definitions of index, nullity and of the different types of min-max families). Theorem 1.1. Let (X, · X ) be a C 2 Finsler manifold modelled on a Banach space E, and Y ֒→ X be a C 2 Finsler-Hilbert manifold modelled on a Hilbert space H which we suppose locally Lipschitz embedded in X, and let F, G ∈ C 2 (X, R + ) be two fixed functions. Define for all σ > 0, F σ = F + σ 2 G ∈ C 2 (X, R + ) and suppose that the following conditions hold.
(2) Energy bound: The following energy bound condition holds : for all σ > 0 and for all {x k } k∈N ⊂ X such that (3) Fredholm property: For all σ > 0 and for all x ∈ K(F σ ), we have x ∈ Y , and the second derivative D 2 F σ (x) : T x X → T * x X restrict on the Hilbert space T x Y such that the linear map is a Fredholm operator, and the embedding T x Y ֒→ T x X is dense for the Finsler norm · X,x . Now, let A (resp. A * , resp. A , resp. A (α * ), resp. A (α * ), where the last two families depend respectively on a homology class α * ∈ H d (Y, B) -where B ⊂ Y is a fixed compact subset -and a cohomology class α * ∈ H d (Y )) be a d-dimensional admissible family of Y (resp. a d-dimension dual family to A , resp. a d-dimensional co-dual family to A , resp. a d-dimensional homological family, resp. a d-dimensional co-homological family) with boundary {C i } i∈I ⊂ Y . Define for all σ > 0 Assuming that the min-max value is non-trivial, i.e.
(4) Non-trivialilty: there exists a sequence {σ k } k∈N ⊂ (0, ∞) such that σ k −→ k→∞ 0, and for all k ∈ N, there exists a critical point x k ∈ K(F σ k ) ∈ E (σ k ) (resp. x * k , x k , x k , x k ∈ E (σ k )) of F σ k satisfying the entropy condition (1. 3) and such that respectively Remark 1.2. The previous theorem is stated for a family F σ = F + σ 2 G, but it would hold more generally under the hypothesis of the previous Theorem ( * ). Notice that the Energy Bound is nothing else that the bound of Theorem ( * ). Firstly, the Palais-Smale condition might be weakened to the Palais-Smale condition along certain near-optimal sequence (see [8]). However, the sequence {σ k } k∈N given by the theorem cannot be made explicit, as is depends on differentiability property of σ → β(σ) (actually, of certain approximations of this function), a function which is a priori impossible to determine explicitly for all σ > 0 (determining β(0) is already a very non-trivial question in many examples, and is actually one of the motivations of the viscosity method), so hypothesis (1) is nearly optimal.
Secondly, the Energy bound is a mere restatement of inequality (1.2), which is really necessary to make the pseudo-gradient argument work (see [15]). It seems to be essentially the only way to obtain Palais-Smale min-max principle.
Thirdly, the restriction on the Hilbert space is used to take advantage of the Morse lemma, a necessary tool in all classical references ( [12], [35], [31], [7], [8]). The Fredholm property is probably necessary as all existing methods rely on perturbation methods using the Sard-Smale theorem ( [29]), for which the Fredholm hypothesis is necessary, thanks of the counter-example of Kupka ([10]). Furthermore, we have to make the hypothesis that T x Y be dense in T x X for a critical point x ∈ K(F σ ) as it shows that the index does not change for the restriction Finally, the Non-triviality assumption is to our knowledge necessary. Indeed, as we cannot localise the critical points of the right index as in the works of Solimini ([31]) and Ghoussoub ([7], [8]), the corresponding theorem is Corollary 10.5 in [7], where this hypothesis is made in order to make sure that one can apply the deformation lemma. Once again, this step is the same that permits to prove the Palais-Smale min-max principle.

Examples of admissible families
We remark that the different families introduced above allow one to recover all known types of min-max considered by Palais in [20]. The only case to check are the homotopy classes of mappings. Let M d be a smooth manifold and let c a regular homotopy class of immersions of Σ k into X, or an isotopy class of embeddings of M d into X. Then is ambient isotopy invariant so is an admissible family of dimension d, i.e. one may freely has additional constraints in the definition of the admissible families as long as they stable under homeomorphisms isotopic to the identity (preserving the boundary conditions, if any). In particular, if Σ k , N n are two smooth manifolds, Imm(Σ k , N n ) is the set of smooth immersions from Σ k to N n , and d ∈ N is such that where π d designs the d-th regular homotopy group, then for all c ∈ π d Imm(Σ k , N n ) with c = 0, and for all l ∈ N and 1 ≤ p < ∞ such that lp > k, as the following Sobolev space of immersion is a smooth Banach manifold ( [23])

Applications
Sacks-Uhlenbeck α-energies ( [28]). Let Σ be a closed Riemann surfaces and let (M n , h) be a closed Riemannian manifold which we suppose isometrically embedded in some Euclidean space R N , and define for all σ ≥ 0 the family of Banach spaces One can check that also X σ depends on σ, as Y is independent of σ, the proof of Theorem 1.1 is still valid. The function F σ : X σ → R is given by where g 0 is some fixed smooth metric on Σ.
The significance of the restriction on the Hilbert space Y is given by the following regularity result (see [17]).
In particular, such critical point u ∈ X σ is an element of Y , and the definition of the index is unchanged, so the main Theorem 1.1 applies.
We find interesting to notice that this idea to restrict a functional defined on a Finsler manifold to a Finsler-Hilbert manifold in order to exploit standard Morse theory in infinite dimension is due to Karen Uhlenbeck ([34]).
In order to introduce the next two categories, we introduce some additional definitions. Let Σ be a closed Riemann surfaces of genus γ, and Diff * + (Σ) be the topological group of positive W 3,2diffeomorphism (we adopt the standard notations of e.g. [3] for Sobolev functions) with either 3 distinct marked points if γ = 0, or 1 marked point for γ = 1 and no mark points for higher genera. Furthermore, if (M n , h) is a fixed Riemannian manifold, we denote by Imm(Σ, M n ) the Banach manifold of W k,pimmersions (for kp > 2) by It was recently proved by Tristan Rivière ([25]) that the quotient spaces X = Imm 2,4 (Σ, M n ) = Imm 2,4 (Σ, M n )/Diff * + (Σ) Y = Imm 3,2 (Σ, M n ) = Imm 3,2 (Σ, M n )/Diff * + (Σ) are (respectively) separated smooth Banach and Hilbert manifolds, and this is really a crucial fact, as by the invariance under the diffeomorphism group on Σ, the perturbed functional of the area of the Willmore energy cannot satisfy the Palais-Smale condition, but satisfies this condition on the quotient space.

Minimal surfaces ([27]
, [21]). Here, the Finsler manifolds are X = Imm 2,4 (Σ, M n ), Y = Imm 3,2 (Σ, M n ) and the functions where g = Φ * h is the pull-back of the metric h on M n by the immersion Φ, and I g is the second fundamental form of the immersion Φ : Σ → M n . However, we see that the subtlety here is that F σ satisfies the Palais-Smale condition only on X, but not on Y . However, as for all critical point Φ ∈ X, we actually have Φ ∈ C ∞ (Σ, M n ), then we have in particular Φ ∈ Y , and one can directly verify that [25]). Therefore, the main Theorem 1.1 applies to the viscosity method for minimal surfaces. Combining the recent resolution of the multiplicity one conjecture proved in this setting by A. Pigati and T. Rivière ([21]) with the previous result of T. Rivière ([25]), one can obtain the lower semi-continuity of the index.

Willmore surfaces ([26], [16]).
The goal here is to go further the minimisation for Willmore surfaces in space forms and to show the existence of Willmore surfaces solution to min-max problems, such as the so-called min-max sphere eversion ( [11]).
Restricting to the special case of Willmore spheres, we take where H g is the mean-curvature of the immersion Φ : S 2 → R n , and O( Φ) is the Onofri energy, defined by where α : S 2 → R is the function given by the Uniformisation Theorem such that g = e 2α g 0 , where g 0 is a fixed metric on S 2 of constant Gauss curvature independent of g. That this quantity is non-negative was proved by Onofri ([18]). Here one also easily proves that the hypothesis of the main Theorem 1.1 are satisfied.
For a proof of the lower semi-continuity of the index and an explicit application, we refer to [14].
Acknowledgements. I would like to thank my advisor Tristan Rivière for his support and very interesting related discussions. I also wish to thank Alessandro Pigati for critically listening a preliminary version of this article.

Organisation of the paper
As is fairly standard in this theory, the proof is divided into two steps between the non-degenerate case and the degenerate case. In the first one, we assume that for all σ > 0, the approximation F σ is non-degenerate and in the second step that ∇ 2 F (x) : T x X → T x X is a Fredholm map at every critical point x ∈ K(F σ ). Through a general perturbation method due to Marino and Prodi ([13]), it is possible to reduce the problem to the non-degenerate case, but this is quite subtle to perturb the function to preserve the entropy condition, contrary to [12] where the degenerate case followed directly from the non-degenerate case.
Furthermore, let us emphasize that there is to our knowledge no method to prove directly Morse index estimates in this setting without reducing to the non-degenerate case, and the Fredholm hypothesis on the second derivative becomes at this point necessary as the only known way to perturb a function on a Finsler-Hilbert manifold to make it non-degenerate is to use the Sard-Smale theorem, for which this hypothesis is necessary.

Preliminary definitions
Definition 2.1. Let π : E → X be a Banach space bundle over a Banach manifold X and let · : E → R + be a continuous function such that for all x ∈ X the restriction · x is a norm on the fibre E x = π −1 ({x}). For all x 0 ∈ X, and for all trivialisation ϕ x0 : We say that · : E → R is a Finsler structure on E is for all x 0 ∈ X and all such trivialisation (U x0 , ϕ x0 ), there exists a constant C = C x0 ≥ 1 such that for all x ∈ U x0 , A Finsler manifold is a regular (in the topological sense) C 1 Banach manifold X equipped with a Finsler structure on the tangent space T X. A Finsler-Hilbert manifold or (infinite-dimensional) Riemannian manifold is a Finsler manifold modelled on a Hilbert space. [20]). Let (X, · ) be a Finsler manifold, and d :

Theorem 2.2 (Palais
Then d is a distance on X inducing the same topology as the manifold topology on X. In particular, we will always assume that Finsler manifolds equipped with their Palais distance, usually denoted by d, and we will denote for all A ⊂ X and δ > 0  (Palais,[20]). Let (X, · ) be a Finsler manifold modelled on some Banach space E, let U ⊂ X be an open subset, ϕ : U → E a chart and x 0 ∈ U . We define for all r > 0 Then for r > 0 sufficiently small B(x 0 , r) is a closed neighbourhood of x 0 , U (x 0 , r) is its interior relative to X and S(x 0 , r) is the frontier relative to X. Corollary 2.4. Let (X, · ) be a Finsler manifold and K ⊂ X be a compact subset. Then for r > 0 small enough, N δ (K) is closed, and U δ (K) is its interior relative to X. Definition 2.5. Let E, F be two Banach spaces. We say that a linear map T ∈ L (E, F ) is a Fredholm operator if Im (T ) ⊂ F is closed, and Ker(T ) ⊂ E and Coker(T ) = F/Im(T ) are finite-dimensional. Then the index Ind(T ) ∈ Z is defined by Definition 2.6. Let X, Y be two Banach manifolds and F : X → Y be a C 1 map. We say that F is a Y is a Fredholm operator and we define the index of F at x, still denoted by Ind x (F ), by Ind x (F ) = Ind(DF (x)).
As the map x → Ind x (F ) ∈ Z is continuous, we deduce that it is constant on each connected component of X, and we will denote it by Ind(F ) if F is defined on a connected domain.
In the applications we have in mind, we cannot assume that the manifold X be connected, so we will have to keep in mind this technical point.

Morse Index and admissible families of min-max
Let X a C 2 Banach manifold, and suppose that F : X → R is a function which admits second order Gâteaux derivatives in X, i.e. for all C 2 path γ : (−ε, ε) → X the function t → F (γ(t)) is a C 2 function. Then a critical point x ∈ X of F is an element such that for all C 2 path γ : (−ε, ε) → X such that If x is a critical point, we define the second derivative quadratic form for all v ∈ T x X and path γ : (−ε, ε) → X such that γ(0) = x and γ ′ (0) = v.
Then Q x is a well-defined continuous map on T x X, and the index Ind F (x) of x with respect to F , is defined by To define the nullity, we need to assume that F : X → R is C 2 Fréchet differentiable map and recalling If F is more regular or X is a Finsler-Hilbert manifold, the definition remains unchanged. That is, if X is a Finsler-Hilbert manifold, then we have for some self-adjoint linear operator L : T x X → T x X. Its number of negative eigenvalues (with multiplicity) is also equal to the index of F by the preceding definition (while the nullity is equal to the number of Jacobi fields, i.e. Null F (x) = dim Ker(L)).

Important remark 2.8.
In particular, if Y ⊂ X is a Lipschitz embedded Hilbert manifold, and x ∈ Y is a critical point of F , then the square gradient ∇ 2 F (x) : T x X → T x X restricts continuously to the Hilbert space T x Y and the definition of the index is unchanged, provided that T x Y ⊂ T x X is dense, a condition easily verified in the cases of interest to us (it is stated explicitly in the hypothesis od the main Theorem 1.1).
We first define families of min-max based on families of continuous maps. Definition 2.9 (Min-max families). Let X be a C 1 Finsler manifold.

(1) Admissible family. We say that
Clearly, this class is stable under homeomorphisms of X isotopic to the identity preserving the boundary h(B).
(2) Dual admissible family. In a dual fashion, let I be a (non-empty) sets of indices and let {C i } i∈I ⊂ X be a collection of subsets such that for all i ∈ I, there exists a non-empty set J i and a family of continuous functions If the functions h i : B d−1 i → X are implicit, then we say by abuse of notation that ) i∈I is the boundary of A (this permits to give a uniform definition of boundary for each of admissible families).
(3) Co-dual admissible family. Finally, given a d-dimension dual admissible family A * , a ddimensional co-dual admissible family is defined by where dim H designs the Hausdorff dimension relative to the Hausdorff measures of the metric space X (equipped with its Palais distance). The class is only stable under locally Lipschitz homeomorphism of X isotopic to the identity (this is not restrictive, as the only homeomorphisms of interest are gradient flow of C 2 functions, which are indeed locally Lipschitz).
Finally, we define the following boundary values of admissible families A , A * and A with boundary

Remark 2.10.
The definition of the third family in [12] is the more restrictive but as we shall see, our definition will still permit to obtain the suitable two-sided index bounds. i i∈I need not be a boundary, but can be any closed subset, as long as it satisfies the non-triviality condition as recalled below. In particular, M d can be assumed to be a cellular complex of dimension at most d. This will be particularly important in the example of Section 3.3, where we shall also in some special situation relax the hypothesis relative to the continuity of the different functions involved in a situation where weaker topologies are available. Definition 2.12. Let X be a C 1 Finsler manifold and A (resp. A * , resp. A ) be a d-dimensional admissible (resp. dual, resp. co-dual) min-max family with boundary {C i } i∈I . We say that A (resp. Whenever this does not yield confusion, we shall write more simply β(A ) and β(A ).
Remark 2.13. The condition (A2) can be relaxed in the sense that the applications f : M d → X need not be continuous with respect to the strong topology of X, as long as we take a weaker notion of continuity stable under homeomorphisms of X isotopic to the identity and fixing the boundary h(B). See Section 3.3 for an explicit example involving families of immersions continuous with respect to the flat norm of currents.
The second class of mappings are based on (co)-homology type properties.
Definition 2.14. Let R be an arbitrary ring, G be an abelian group, and d ∈ N a fixed integer.
is the induced map in cohomology from the injection ι A : A → X. In other word, the non-zero class α * is not annihilated by the restriction map in cohomology ι *

Remark 2.15. This recovers the classes (e) and (f) in the seminal paper of Palais ([19]
). We observe that for cohomological families, there is no boundary conditions to check, as they are obviously stable under any ambient homeomorphism isotopic to the identity Id X : X → X. One can check that no restrictions is necessary for the coefficients in homology and cohomology.

Deformation lemmas
The results we present here are essentially adaptations to our setting of known results of Lazer-Solimini and Solimini (see also the results of Ghoussoub for subsequent extensions [7], [8]).
The next lemma is due to Solimini and absolutely crucial as, whereas the restriction of F σ on the Hilbert does not satisfy the Palais-Smale condition, it satisfies a stronger property on a suitable neighbourhood of critical points.
If (X, · ) is a Finsler manifold equipped with its Palais distance d and A ⊂ X, we recall the notations Notice in particular that by Corollary 2.4, if A is assumed to be compact, then N δ (A) is closed and U δ (A) is its interior. In all constructions, we will assume implicitly whenever necessary that such δ > 0 has been chosen such that N δ (A) is closed. (Solimini,[31]). Let X be a C 2 Finsler-Hilbert manifold and F : X → R be a C 2 function, and assume that

Proposition 2.16
In particular, F satisfies the Palais-Smale condition on N δ (K).

Remark 2.17.
That DF is proper near critical points x ∈ X where D 2 F (x) is Fredholm is a well-known property due to Smale ([30]).
Proof. We first treat the case K = {x 0 }. By a remark which will be repeatedly used, we can assume by Henderson's theorem ( [9]) that X is a open subset of a Hilbert space H. We fix some ε > 0, and we take and define G : X → H by Then G is Lipschitz and satisfies by (2.2) In particular, as H 0 is finite dimensional, there exists a positive constant 0 < α < ∞ such that for all v ∈ H ⊥ 0 , there holds according to the direct sum decomposition (2.4), we can assume that up to subsequence x 0 k k∈N is convergent in N δ (x 0 ) (which is closed by Corollary 2.4). Now, for all k, l ∈ N, we have by the assumption and the previous remark. This finishes the proof of the special case of the proposition.
As K is compact, there exists a uniform α such that (2.5) holds for all x 0 ∈ K and appropriate H 0 = H 0 (x 0 ). Taking a finite covering {N δ (x i )} 1≤i≤N for δ > 0 small enough and some elements {x i } 1≤i≤N ⊂ K, the previous proof works identically. This concludes the proof of the general case.
Proof. As K is compact, let x 1 , · · · , x n ∈ K be such that Taking δ small enough, we can make sure that each ball and let ϕ i ∈ C ∞ (X, R) be defined by has all the required properties.
We recall the proof of the following perturbation method due to Marino and Prodi, as we will have to exploit the specific form of the perturbation in the proof of the main Theorem 1.1. Proposition 2.20 (Teorema 2.1 [13], Proposition 3.4 [31]). Let k ≥ 2 and X be a C k Finsler-Hilbert manifold and F : X → R be a C k function, and assume that and the critical points of F in N δ (K) are non-degenerate and finite in number. Furthermore, we can Proof. We can assume by Henderson's theorem that X is an open subset of a Hilbert space H with scalar product · , · . Let ϕ : X → R be the cut-off function of Lemma 2.19, and define for x 0 ∈ N 2δ (K 0 ), y ∈ X the function F x0,y : X → R such that for all x ∈ X, Furthermore, thanks of the construction of Lemma 2.19, we have for some universal constant C 1 Then for all y ≤ δ k C0(δ)C1 ε, we get the the first property of (2.6). Furthermore, we have on N δ (K) In particular, x ∈ N δ is a critical point of F if −y is a regular value of ∇ F x0,y : X → H. Now, if we take δ > 0 small enough such that each connected component of N δ (K) intersects K, we see that DF is a Fredholm map on N δ (K) of index 0. Indeed, as H is a Hilbert space, seeing all x ∈ N δ (K) as a is a self-adjoint Fredholm operator (by the connectedness hypothesis), so it must have index 0. In particular, we can apply the Sard-Smale Theorem 2.7 if ∇F is only C 1 on X to obtain an element −y ∈ X such that which is a regular value of ∇F x0,y (for all x 0 ∈ X). Writing F x0 = F x0,y , we see that for all x 0 ∈ N 2δ (K), by (2.7), (2.8) and (2.9) and their exists x 0 ∈ N 2δ (K) such that Taking F = F x0,y , we obtain F ≤ F and the conclusions of the Proposition (the other inequality F ≥ F is similar).

Deformation and extension lemmas
As a key technical lemma in [12] contains an incorrect statement, we will check in this section that Lazer-Solimini's construction does not actually use this statement, so that their results are still valid (along with [31]).
As we have mentioned it earlier, the basic principle to obtain index bounds is to first consider the case of non-degenerate functions. Therefore, we fix a C 2 Finsler-Hilbert manifold X (modelled on a separated Hilbert) and a C 2 function F : X → R, for which we assume that F satisfies the Palais-Smale condition at all level c ∈ R, and to fix ideas, let A be a d-dimensional admissible family. We assume that F is non-degenerate on the critical set K(F, β 0 ) at level β 0 = β(F, A ). In particular, as F satisfies the Palais-Smale condition, K(F, β 0 ) is compact and as F is non-degenerate on K(F, β 0 ), we deduce that K(F, β 0 ) is composed of finitely many points, so that for some x 1 , · · · , x m ∈ X, we have Let i ∈ {1, · · · , m} be a fixed integer. Then there exists closed subspaces where · is the norm of the Hilbert space H. In order to make the notations lighter, we will remove most explicit dependence in the index i in the following of the presentation. Now, we let r 1 , r 2 > 0 be such that 2r 1 < r 2 and small enough such that the closed balls B − (0, r 1 ) ⊂ H and B + (0, r 2 ) ⊂ H such that B − (0, 2r 1 ) + B + (0, r 2 ) ⊂ ϕ(U εi (x i )). Now, we define for all 0 < s ≤ 2r 1 and 0 < t ≤ r 2 Now, fix 0 < δ < r 2 2 − 4r 2 1 , and let ζ : R → [0, 1] be a smooth cut-off function such that supp(ζ) ⊂ R + and ζ(t) = 1 for all t ≥ 1. Now, we define a map Φ : X → X such that

Remark 3.2.
It is also claimed (without proof, which is left to the reader) in [12] and [8] that we have the additional property: (3) We have Φ(X \ C(r 1 , r 2 )) ⊂ X \ C(r 1 , r 2 ).
This implies that Φ(ϕ −1 (B − (0, δ(2r 1 )) + ∂B + (0, r 2 ))) ⊂ int(C(r 1 , r 2 )) and as trivially ϕ −1 (B − (0, δ(2r 1 )) + ∂B + (0, r 2 )) ⊂ int(C(r 1 , r 2 )) = ϕ −1 (U − (0, r 1 ) + U + (0, r 2 )) , (3.4) we see that property (3) is actually false (as the set on the left-hand side of (3.4) is non-empty). However, it does not enter in the proof of the main theorem in [12], as we shall see below. Proof. First assume that K is compact, and let r > 0 such that K ⊂ B(0, r). Then we obtain an extension f : M d → H by a theorem of Dugundji (see [5]) through partition of unity. Furthermore, as M d is a smooth manifold, we can take the partition of unity to be C 1 so that the restriction f | M d \K : If K is not compact, we fix some arbitrary point p ∈ M d and for all n ∈ N, we let K n = K ∩ B(p, n). We apply the previous construction to the restriction f K1 :  , so we could relax the hypothesis to metric spaces admitting α-Hölder partitions of unity.

The index bounds for non-degenerate functions on Finsler-Hilbert manifolds
Definition 3.5. If A is a min-max family and F ∈ C 1 (X, R), and we define Theorem 3.6 (Lazer-Solimini, [12]). Let X be a C 2 Finsler-Hilbert manifold, A (resp. A * , resp. A ) be a d-dimensional admissible family (resp. dual family, resp. co-dual family) with boundary {C i } i∈I ⊂ X and let F ∈ C 2 (X, R) be such that F satisfies the Palais-Smale at level β 0 = β(F, A ). Assume furthermore that all critical points of F are non-degenerate at level β 0 , and that the min-max is nontrivial, i.e.

Case 2: dual admissible families.
In this case, the construction is straightforward, as we will show that under the same notations for the Morse transformation, we have for all 0 < η < δ and for all A ∈ A * such that there holds (notice that Φ(A) ∈ A * by construction of Φ) which will immediately imply the claim, as F (Φ(x)) ≤ F (x) for all x ∈ X, so that Now assume by contradiction that (3.12) does not hold. This means by Definition 2.9 that there exists a continuous map h : Φ(A) \ int(C(r 1 , r 2 )) → R d \ {0} such that for some i ∈ I and j ∈ J i , we have

we deduce by Lemma 3.1 that there exists an extension
is the continuous map given by this implies by definition of A * that Φ(A) / ∈ A * , a contradiction (as 0 / ∈ Im( h)).

Case 3: co-dual admissible families.
First, the argument of Case 2 shows that we only need to treat the case Ind F (x 0 ) < d, as the map ϕ : U ε (x 0 ) → ϕ(U ε (x 0 )) ⊂ H − is a locally bi-Lipschitz homeomorphism, so the map Φ : X → X is locally Lipschitz on A, so that dim H (Φ(A)) ≤ dim H (A) < d + 1 (3.13) and as Φ(A) ∈ A * , we obtain by (3.13) that Φ(A) ∈ A .
Proof. (of Theorem 3.6) As the conclusions of Proposition 3.8 are independent of the admissible family, we can assume that {A k } k∈N ⊂ X is such that A k ∈ A for all k ∈ N and

Now, let
Then by assumption, F is non-degenerate on K(F, β 0 ) ∩ A ∞ , and as K(F, β 0 ) ∩ A ∞ is compact by the Palais-Smale condition, we deduce by the Morse lemma that K(F, β 0 ) ∩ A ∞ is finite, so we have for some Now, we taking ε > 0 sufficiently small, we can assume that Thanks of (3.19) and (3.20), we see that A k satisfies the hypothesis to obtain (3.19) for k large enough define by a finite induction A 1 k , · · · A m k ∈ A by so by any deformation lemma (see e.g. [31] β 0 ). Furthermore, assuming that ε > 0 is small enough, and as {x 1 , · · · , x m } = which furnishes the desired contradiction. Proposition 3.9. Let d ≥ 1 be a fixed integer, R be an arbitrary ring, G be an abelian group, F ∈ C 2 (X, R + ) as in Theorem 3.6, B ⊂ X a compact subset, α * ∈ H d (X, B, R) \ {0} and α * ∈ H d (X, G) \ {0} be non-trivial classes in relative homology and cohomology respectively, let A (α * ) and A * be the corresponding d-dimensional homological and cohomological admissible families, and be the associated width. Assume that x 0 ∈ K(F, β 0 ) (resp. x 0 ∈ K(F, β 0 )) is a non-degenerate critical points of F at level β 0 (resp. β 0 ) and that Then for all small enough ε > 0, there exists δ > 0 such that for all A ∈ A (α * ) (resp. A (α * )), Proof. Let x 0 ∈ K(F, β 0 ) be a non-degenerate critical critical point, and let r 1 , r 2 , δ > 0 be given by Lemma 3.1 such that 0 < δ < r 2 2 − 4r 2 1 , and A ∈ A (α * ) such that Then by definition, α ∈ Im(ι A, * ), where ι A, * : H d (A, B) → H d (X, B) is the induced map in relative homology from the inclusion ι A : A → X. We will now show that for all 1/2 < ε < 1 close enough to 1, we have εr 1 , 0))) ∈ A (α * ).

By (3.25) and (3.26), we obtain
and we obtain the following exact sequence and as Im(f ) = Ker(g) = H d (Φ(Y )), we deduce that f is surjective. Now, as the map Φ : X → X given by Lemma 3.1 is continuous on X ∩ {x : F (x) ≤ β 0 + δ} and isotopic to the identity on X ∩ {x : F (x) ≤ β 0 + δ} (which contains Y ), we deduce that the Φ * homomorphisms in the Mayer-Vietoris commutative diagram are isomorphism, so we have a surjection In particular, the arrow h of the following we obtain a surjection Now, if B ⊂ A 1 ⊂ A 2 ⊂ X are any two subsets containing B, we write ι A1,A2 : A 1 ֒→ A 2 the injection and ι A1,A2 * : and as α * ∈ Im(ι A,X, * ) ⊂ H d (X, B), this implies that α * ∈ Im(ι Y,X * ), and by the surjectivity of the arrow in (3.27), we obtain r1,ε r2)),X * , which by definition means that (notice that Y is compact) Finally, for all x ∈ C(r 1 , 0) \ int(C(εr 1 , 0)), we have Using the exact same arguments of proof (with A ′ = A \ int(C(ε r 1 , ε r 2 )) ∪ C(r 1 , 0) \ int(C(ε r 1 , 0))) thanks of the Mayer-Vietoris sequence for singular cohomology, we show the injectivity of the following arrow εr 1 , 0)), G) and this finishes the proof of the theorem.

Remark 3.10.
We see that there is absolutely no restriction in the coefficients in (singular) homology of cohomology, as we only used Mayer-Vietoris exact sequence.

Corollary 3.11.
Under the hypothesis of Proposition 3.9, if F ∈ C 2 (X, R) and (3.28) Proof. It is exactly the same as the proof of Theorem 3.6, using Proposition 3.9 instead of Proposition 3.8.

Application to the min-max hierarchies for minimal surfaces
We observe that the previously considered admissible families need not be continuous with respect to the strong topology on X, as the following corollary shows. This application is of interest in the setting of min-max hierarchies for minimal surfaces recently developed by Tristan Rivière ([25]). We first introduce some terminology (see [6]  Let Σ be a closed Riemann surface, N n be a compact Riemannian manifold with boundary (possibly empty) which we suppose isometrically embedded in some Euclidean space, and G 2 (T N n ) be the Grassmannian bundle of oriented 2-planes in T N n . We denote by V 2 (N n ) the space of 2-dimensional varifolds on N n , that is the space of Radon measure on G 2 (T N n ) endowed with the weak- * topology. Furthermore, we denote by Z 2 (N n , G) the space of rectifiable 2-cycles in N n with G-coefficients (see [6], 4.1.24, 4.2.26, 4.4.1), where G = Z or G = Z 2 (or more generally, G is an admissible in Almgren's sense [2]). It is known that every current T ∈ Z k (N n , G) induces a varifold |T | ∈ V 2 (N n ), and we denote by F the flat norm on Z 2 (N n , G) and by d V the varifold distance, defined for all V, W ∈ V 2 (N n ) by Furthermore, if Φ ∈ Imm 3,2 (Σ, N n ) is a W 3,2 immersion as defined in Section 1.2, then obviously the push-forward Φ * [Σ] of the current of integration [Σ] on the closed Riemann surface Σ is an element of Z 2 (N n , Z), and furthermore, the induced varifold is denoted by We introduce the following distance on V 2 (N n ) ∩ {|T | : T ∈ Z 2 (N n , G)}: for all V, W ∈ V 2 (N n ) such that V = |S| and W = |T | for some S, T ∈ Z 2 (N n , G), Finally, if for all g ∈ N, Σ g is a fixed closed oriented surface of genus g, we denote by Imm 0 3,2 (Σ g , N n ) the connected component (for regular homotopy) of the immersions regularly homotopic to an embedding Σ g ֒→ N n , on we denote by Imm ≤g0 (N n ) the disjoint union of Finsler-Hilbert manifolds We introduce for all 0 ≤ σ ≤ 1 the function A σ : Imm ≤g0 (N n ) → R defined for all Φ ∈ Imm ≤g0 (N n ) by if Φ is defined from a closed surface Σ, and I Φ is its second fundamental form. That A σ satisfies all hypothesis of Theorem 1.1 is verified in [27].
and define for all 0 ≤ σ ≤ 1 Assuming that A is non-trivial as in Theorem 1.1, there exists a sequence {σ k } k∈N ⊂ (0, ∞) such that σ k → 0 and and for all k ∈ N, there exists a critical point Proof. As the extensions are made for maps whose domains and co-domains is finite-dimensional, by the equivalence of norms in finite dimension, the different restriction of the sweep-outs are continuous in any topology, and the extension can be taken Lipschitz in the strong topology on W 3,2 immersions, so the proof is virtually unchanged.

The entropy condition
Let X be a Finsler manifold and {F σ } σ∈[0,1] ⊂ C 1 (X, R) such that for all x ∈ X, σ → F σ (x) is increasing. If A is any of the admissible families, we define for all σ ∈ [0, 1] As the function σ → β(σ) is increasing, it is differentiable almost everywhere (with respect to the 1-dimensional Lebesgue measure) and we have Suppose by contradiction that this is not the case. Then there exists δ > 0 such that for σ > 0 small enough which contradicts (4.1).

Definition 4.1.
We say that β satisfies the entropy condition at σ > 0 if β is differentiable at σ and if In particular, there always exists a sequence of positive number {σ k } k∈N such that σ k −→ k→∞ 0 and β verifies the entropy condition at σ k .

The non-degenerate case
If X is a Finsler-Hilbert manifold and F : The next result is a variant of [31], 2.13 [8], 4.5, which will allow us to construct critical points of the right index. It permits to show that we can always obtain the entropy condition as we locate critical points in some almost critical sequence.

Theorem 4.2.
Let X be a Banach manifold and F, G ∈ C 2 (X, R + ), A an admissible min-max family, and define for 0 ≤ σ < 1 the function F σ = F + σ 2 G, and and assume that the Energy bound (2) of Theorem 1.1 holds. Now suppose that β is differentiable at 0 < σ < 1 and satisfies the entropy condition, i.e. . .
Proof. Looking at Step 2 of the proof of Proposition 6.3 (it is written for geodesics, but the same proof work equally well in general, see [24]), we see that assuming by contradiction that for all for k ≥ 1 large enough, we have for all x ∈ X such that dist(x, A k ) ≤ δ k and then for some δ k > 0 to be determined later, there exists a semi-flow {ϕ t k } t≥0 : X → X isotopic to the identity and preserving the boundary of A such that for all 0 ≤ t ≤ δ k (as dist(x, ϕ t (x)) ≤ t for all t ≥ 0), and x ∈ A k such that (4.2) is satisfied, there holds In particular, as ϕ t k (A) ∈ A , we have so we deduce that for all 0 ≤ t ≤ δ k by (4.3) Furthermore, as β is differentiable at σ, we can assume that k is large enough such that so by (4.4) and (4.5), we have for t = δ k and η k = ϕ t k : Therefore, choosing a contradiction. Therefore, we see that there exists x k ∈ X such that where the last condition is given by the identity below (6.11) in [15]. Finally, this is easy to see that (3) ′ implies the (3) of the theorem (thanks of the Energy bound condition), and this concludes the proof (see [27] for the optimal hypothesis on F σ for this assertion to hold true).

Remark 4.3.
If Y ֒→ X is a locally Lipschitz embedded Hilbert-Finsler manifold, and A ⊂ P(Y ) is an admissible family (i.e. it is stable under locally Lipschitz homeomorphisms of Y ), then the restriction F |Y is still C 2 and by taking pseudo-gradients with respect to this restriction, we see that any A ∈ A will be preserved by the map ϕ t δ k . Therefore, we obtain a sequence {x k } k∈N ⊂ Y such that (4.6) are satisfied with respect to the Finsler norm and distance on Y , and by the local Lipschitz embedding, we also obtain dist X (x k , A k ) −→ k→∞ 0, and DF σ k X,x k −→ k→∞ 0. Using the Palais-Smale condition and the energy bound valid with respect to X, the end of the proof is identical.
. Theorem 4.5. Let X be a C 2 Finsler manifold, F, G ∈ C 2 (X, R + ), and define for all σ ≥ 0 the function F σ = F + σ 2 G ∈ C 2 (X, R + ) and let A (resp. A * , resp. A ) be a d-dimensional admissible family (resp. a dual family, resp. a co-dual family). Assume that F σ satisfied the hypothesis of Theorem 1.1, and let

be a min-maximising sequence such that
and assume that all critical points of Remark 4.6. Likewise, the proof would work equally well for homotopical and cohomotopical families, by Proposition 3.9.
Proof. We give the proof in the special case where X is C 3 and F, G ∈ C 3 (X, R), in order to use Morse lemma as in [19]. However, as the extension of the Morse lemma to C 2 spaces and functions ( [4]) is based on Cauchy-Lipschitz theorem and by the continuous dependence at the existence time with respect to the flow, the proof given below readily generalises to this weaker setting.
. As the critical points in K are non-degenerate, K is compact and consists of finitely many points {x 0 , x 1 , · · · , x m } ⊂ K β(σ) . We cannot apply the previous lemma on F σ as the main lemma only work with F σ k . First by the Palais-Smale condition for F σ and as the critical points are isolated, we deduce that there exists δ > 0 such that B 2δ (x i ) ∩ B 2δ (x j ) = ∅ for all i = j and Also notice that thanks of the proof of Theorem 4.2, for all so up to a subsequence, we have thanks of the Palais-Smale condition for F σ at level β(σ) that x k −→ k→∞ x ∈ K β(σ) . In particular, if {x k } ⊂ X verifies (4.8), then we can assume up to some relabelling that that for all k ∈ N large enough x k ∈ N δ (x 0 ). Now, looking at the proof of Morse Lemma by Palais ([19]) which only works for C 3 functions, we see that the diffeomorphism ϕ around a critical point x i such that where for all v, w ∈ H, we have by Taylor expansion for some map A x0 : B(x 0 , δ) → L (H) with values into self-adjoint continuous operators we deduce that for some δ 0 > 0 small enough and depending only on A x0 , namely such that for all that ϕ(x) is well-defined on B(x 0 , δ) and C 1 . Therefore, thanks of the local inversion theorem, up to diminishing δ, we can assume that ϕ is a diffeomorphism from B(x 0 , δ) onto its image (here, δ depends only on A x0 ).
x 0 . Thanks of the strong convergence, we deduce that for k large enough, In particular, if k is large enough such that we deduce by (4.9) that In particular, we can define ϕ x k (x) = B x k (x)x for all x ∈ B(x, δ 2 ), and we see that in particular dϕ k (x k ) = Id. Now, as and as the neighbourhood around which ϕ x k is invertible depends only on the local behaviour of its derivative around x k and as ϕ x0 is invertible in B(x 0 , δ), we deduce that for k large enough, ϕ k is invertible on B(x k , δ 4 ), so the Morse lemma implies that In particular, F σ k has only one critical point on B(x 0 , δ 8 ) ⊂ B(x k , δ 4 ) for k large enough. Therefore, we can apply the Proposition 3.8 to F σ k with δ > 0 and ε > 0 independent of k.
As K = {x 0 , x 1 , · · · , x m } is finite, we saw that for all k sufficiently large, F σ k has at most one critical where m k ≤ m the critical points of F σ k at level β(σ k ). Thanks of Proposition 3.8 and the first part of the proof, there exists some δ > 0 independent of k such that for all A ∈ A such that sup F σ k (A) ≤ β(σ) + δ, then for all (4.10) Furthermore, as the x k i are uniformly isolated independently of k, taking ε > 0 small enough, we can assume that and that if x k i −→ k→∞ x j ∈ K (for some j ∈ {1, · · · , m}) that k is large enough such that . We also remark that is an open neighbourhood of In particular, there exists k 0 ∈ N such that for all k ≥ k 0 , there holds We can also assume as K is isolated in K β(σ) ∩ E (σ) and thanks of the first part of the proof that ε > 0 is small enough such that using the notation of (4.10). We see in particular that by (4.10) and (4.11) (4.14) Furthermore, as for all 1 ≤ j ≤ m k , we have by (4.10) so by combining (4.13), (4.12) with (4.14) and (4.15), we deduce that for all k ≥ k 0 , we have (4.18) Therefore, by the Palais-Smale condition at level β(σ) and (4.18), up to a subsequence we have x k −→ k→∞ x ∞ ∈ K β(σ) ∩ E (σ). However, we have for all k large enough by (4.16) and as dist( and this contradicts the fact that x ∞ ∈ K β(σ) ∩ E (σ). This concludes the proof of the theorem.

Marino-Prodi perturbation method and the degenerate case
Let us recall the main theorem here.
Theorem 4.7. Let (X, · X ) be a C 2 Finsler manifold modelled on a Banach space E, and Y ֒→ X be a C 2 Finsler-Hilbert manifold modelled on a Hilbert space H which we suppose locally Lipschitz embedded in X, and let F, G ∈ C 2 (X, R + ) be two fixed functions. Define for all σ > 0, F σ = F + σ 2 G ∈ C 2 (X, R + ) and suppose that the following conditions hold.
(2) Energy bound: The following energy bound condition holds : for all σ > 0 and for all {x k } k∈N ⊂ X such that (3) Fredholm property: For all σ > 0 and for all x ∈ K(F σ ), we have x ∈ Y , and the second derivative is a Fredholm operator, and the embedding T x Y ֒→ T x X is dense for the Finsler norm · X,x . Now, let A (resp. A * , resp. A , resp. A (α * ), resp. A (α * ), where the last two families depend respectively on a homology class α * ∈ H d (Y, B) -where B ⊂ Y is a fixed compact subset -and a cohomology class α * ∈ H d (Y )) be a d-dimensional admissible family of Y (resp. a d-dimension dual family to A , resp. a d-dimensional co-dual family to A , resp. a d-dimensional homological family, resp. a d-dimensional co-homological family) with boundary Assuming that the min-max value is non-trivial, i.e.
(4) Non-trivialilty: there exists a sequence {σ k } k∈N ⊂ (0, ∞) such that σ k −→ k→∞ 0, and for all k ∈ N, there exists a critical point x k ∈ K(F σ k ) ∈ E (σ k ) (resp. x * k , x k , x k , x k ∈ E (σ k )) of F σ k satisfying the entropy condition (1.3) and such that respectively Proof. As we have mentioned already, we can assume that X is a Finsler-Hilbert manifold modelled on a Hilbert space H. Take σ > 0 such that β satisfies the entropy condition at σ. If F σ has only non-degenerate critical points in K β(σ) ∩ E (σ), then we are done. Proof. By contradiction, let δ > 0 such that Then there exists J ∈ N such that for all j ≥ J, so that for all j ≥ J, there holds δ b j ≤ a j . Therefore, we obtain contradicting the divergence of b j .
Therefore, if b = {b j } j∈N is a the general term of a divergent series with positive terms, there exists by Lemma 4.8 a subsequence {j l } l∈N such that for all l ∈ N, there holds Now, for convenience of notation, as we do not use any properties related to the convergence of the series of general term {b j l } l∈N , we will assume that (4.19) holds for all j ∈ N. Now, we want to find such sequence {a j } j∈N and {b j } j∈N such that (a j − a j+1 ) −1 b j ≤ 1 a j log 1 aj log log 1 aj log log log 1 aj Take a j = 1 j , we have a j − a j+1 = 1 j(j + 1) , so the condition becomes for j ≥ 4 · 10 6 > e e e b j ≤ 1 (j + 1) log(j) log log(j) log log log(j) , (4.20) and the series whose general term is the right-hand side of (4.20) diverges so we define {b j } j∈N ⊂ (0, ∞) N such that for all j ≥ J ≥ 4 · 10 6 > e e e b j = 1 (j + 1) log(j) log log(j) log log log(j) .
Now, for all j ≥ J, let I j = [a j+1 , a j ] and and define δ j for j ≥ J by Then for all j ≥ J, there holds by (4.19) aj aj+1 β ′ (σ)dσ ≤ δ j (a j − a j+1 ) a j log( 1 aj ) log log( 1 aj ) so that a j log 1 aj log log 1 aj so that Therefore, we obtain for all j ≥ J some element σ j ∈ (a j+1 , a j ) such that . Now, for all σ ∈ (0, 1), as K(F σ ) is compact, we let ϕ σ be the cut-off function given by Proposition 2.20 and let ε(σ) > 0 such that for all y < ε(σ) small enough such that by Proposition 2.16, the map F σ,y = F σ + ϕ σ y, · (4.22) is proper on N 2δ (K). Now, fix some C > 0.
First, observe that F σ,y has no critical points in X \ K(F σ ) 2δ , as F σ,y = F σ in X \ K(F σ ) 2δ . Now, by contradiction, assume that there exists {τ k } k∈N such that τ k → σ and a sequence of critical points {x k } k∈N ⊂ X (i.e. such that x k ∈ K(F τ k σ,y ) ∩ {x : F τ σ (x) ≤ C} for all k ∈ N) and dist(x k , K(F σ )) ≥ δ.
Then, by the same proof mutadis mutandis of (6.9) of Proposition 6.3 in [15], we have thanks of the condition (2) on the energy bound that for k large enough, as ∇F σ,y is proper on K(F σ ) 2δ , we deduce that up to a subsequence, we have a contradiction. Therefore, we can assume that for all k ∈ N) and dist(x k , K(F σ )) ≥ 2δ.
Furthermore, as F τ k σ,y = F τ k and F σ,y = F σ on X \ K(F σ ) 2δ , we have Therefore, by the Palais-Smale condition for F σ , we deduce that up to subsequence, we have x k −→ k→∞ x ∞ ∈ K(F σ ), a contradiction. Now, to prove the second part of the claim, by (4.23), if τ k −→ k→∞ σ and {x k } k∈N ⊂ X is a sequence of critical points associated to F τ k σ,y k∈N such that we have by properness of F σ,y on K(F σ ) 2δ that (up to a subsequence) x k −→ k→∞ x ∞ ∈ K(F σ,y ). Furthermore, the strong convergence of {x k } k∈N towards x ∞ shows that as we see these two second order operators defined on the underlying Hilbert space H ≃ T x k X ≃ T x∞ X. Now, we recall the following continuity property of the spectrum for bounded linear operators on a Hilbert space H, which we state below.

σ(y)
, and Ind F σ(y),y (x(y)) ≤ d. (4.27) As the set of y ∈ X such that F σ,y is non-degenerate is dense, we can choose a sequence {y k } k∈N ⊂ X such that y k −→ k→∞ 0, such that F σi,y k is non-degenerate for all 1 ≤ i ≤ N j for all k ∈ N, and σ(y k ) = σ j k ∈ I j such that F σ k ,y k admits a critical point x(y k ) = x k ∈ X verifying (4.27). As I j is compact, we can assume up to a subsequence that σ j k −→ k→∞ σ j ∞ ∈ I j , and as we deduce that up to a subsequence, by the Energy bound (2) and (4.28), we have (notice that ∇F σ k ,y k (x k ) = 0) Therefore, up to an additional subsequence and by the Palais-Smale condition, we have the strong convergence . Finally, by the strong convergence of the second derivative, we have Ind Fσ k ,y k (x k ) (4.29) and (notice that by non-degeneracy of x k for F σ k ,y k that Null Fσ k ,y k (x k ) = 0) Ind Fσ k ,y k (x k ) + Null Fσ k ,y k (x k ) (4.30) so x j ∞ verifies (4.27) for y = 0 and σ(y) = σ j ∞ . Furthermore, if A is replaced by a dual family, then the one-sided estimate from below of the index is given by (4.30) while two sided estimates are given for co-dual, homological or cohomological families by (4.29) and (4.30).
This concludes the proof of the theorem, as the sequences σ j ∞ j∈N ⊂ (0, ∞) and x j ∞ j∈N ⊂ X satisfy the conditions of the theorem.