On the Morse index of critical points in the viscosity method

Abstract

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.

1 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 $F:X\rightarrow {\mathbb {R}}$, for which one aims at constructing (unstable) critical points. We further fix some d-dimensional compact manifold $M^d$ with boundary $\partial M^d= B^{d-1}\ne \varnothing$, and a continuous map $h: B^{d-1}\rightarrow X$, and we call the subset ${\mathscr {A}}\subset {\mathscr {P}}(X)$ a d-dimensional admissible family (relative to $(M^d,h)$) if

$$\begin{aligned} {\mathscr {A}}&={\mathscr {P}}(X)\cap \big \{A: \text {there exists a continuous map}\; f:M^d\rightarrow X\; \text {such that}\; A=f(M^d)\\&\quad \text {and}\; f_{B^{d-1}}=h\big \}. \end{aligned}$$

We shall generalise this example later and introduce additional min–max families in Sect. 2.2. First recall the definition of the critical set $K(F)\subset X$ of critical points of a function $F:X\rightarrow {\mathbb {R}}$:

$$\begin{aligned} K(F)=X\cap {\left\{ x:DF(x)=0\right\} } \end{aligned}$$

In particular, notice that ${\mathscr {A}}$ is stable under homeomorphisms isotopic to the identity preserving the boundary $h(B^{d-1})\subset X$. Then the min–max level associated to F and ${\mathscr {A}}$, denoted here by $\beta (F,{\mathscr {A}})$ or ($\beta ({\mathscr {A}})$ when there is no ambiguity in the choice of F) is defined by

$$\begin{aligned} \beta (F,{\mathscr {A}})=\inf _{A\in {\mathscr {A}}}\sup F(A)<\infty . \end{aligned}$$

Assuming that the min–max is non-trivial in the following sense

$$\begin{aligned} \beta ({\mathscr {A}})=\inf _{A\in {\mathscr {A}}}\sup F(A)>\sup F(h(B))={\widehat{\beta }}({\mathscr {A}}), \end{aligned}$$

this is a very classical theorem of Palais [19] that there exists a critical point $x\in K(F)$ of F such that $F(x)=\beta ({\mathscr {A}})$, 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 $\nabla ^2 F(x):T_xX\rightarrow T_xX$ is a Fredholm operator at every critical point $x\in K(F)$. We also define the index $\mathrm {Ind}_F(x)\in {\mathbb {N}}$ (resp. the nullity $\mathrm {Null}_F(x)$) of a critical point $x\in K(F)$ of F as the number (with multiplicity) of negative eigenvalues (resp. as the multiplicity of the 0-eigenvalue) of the Fredholm operator $\nabla ^2 F(x):T_xX\rightarrow T_xX$.

In this setting, it was subsequently proved by Lazer and Solimini [12] that it is possible to find a critical point $x^{*}\in K(F)$ (a priori different from x) such that the following index bound holds

$$\begin{aligned} \mathrm {Ind}_{F}(x^{*})\le d. \end{aligned}$$

(1.1)

For different types of min–max family, it is also possible to obtain a one-sided bound

$$\begin{aligned} \mathrm {Ind}_{F}(x^{*})+\mathrm {Null}_{F}(x^{*})\ge d \end{aligned}$$

or a two-sided estimate

$$\begin{aligned} \mathrm {Ind}_{F}(x^{*})\le d\le \mathrm {Ind}_{F}(x^{*})+\mathrm {Null}_{F}(x^{*}). \end{aligned}$$

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 Sect. 2.2). For min–max families defined with respect to homology classes, the two-sided estimate was first obtained by Viterbo [38].

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 (classical examples are given by 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 $G:X\rightarrow {\mathbb {R}}_+$ be a $C^2$ function and we define for all $\sigma >0$ the $C^2$ function $F_{\sigma }=F+\sigma ^2G$, and we assume that for all $\sigma >0$, the function $F_{\sigma }:X\rightarrow {\mathbb {R}}$ verifies the Palais–Smale condition. Furthermore, we denote for all $\sigma \ge 0$ (so that $\beta (0)=\beta ({\mathscr {A}})$)

$$\begin{aligned} \beta (\sigma )=\beta (F_{\sigma },{\mathscr {A}})\ge \beta ({\mathscr {A}})=\beta (0). \end{aligned}$$

In particular, the previous theory applies and we can find for all $\sigma >0$ a critical point $x_{\sigma }$ of $F_{\sigma }$ of the right index. Then this is a case-by-case analysis to show that the bounds carry as $\sigma \rightarrow 0$ (see [30] 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 ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}\subset (0,\infty )$ converging towards 0 and some sequence ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ of critical points associated to ${\left\{ F_{\sigma _k}\right\} }_{k\in {\mathbb {N}}}$ (i.e. such that $x_k\in K(F_{\sigma _k},\beta (\sigma _k))$ for all $k\in {\mathbb {N}}$)

$$\begin{aligned} F_{\sigma _k}(x_k)=\beta (\sigma _k){\underset{k\rightarrow 0}{\longrightarrow }}\beta (0)=\beta ({\mathscr {A}}),\quad \text {and}\quad \limsup _{k\rightarrow \infty }F(x_{\sigma _k})<\beta ({\mathscr {A}}). \end{aligned}$$

There are some explicit examples of such failures (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 [35, 36]). In our setting, the corresponding theorem is the following (see [15] or [25] for a proof).

Theorem

($*$) Let $(X,\Vert \,\cdot \,\Vert )$ be a complete $C^1$ Finsler manifold. Let $F_{\sigma }:X\rightarrow {\mathbb {R}}$ be a family of $C^1$ functions for all $\sigma \in [0,1]$ such that for all $x\in X$,

$$\begin{aligned} \sigma \mapsto F_{\sigma }(x) \end{aligned}$$

is $C^1$ and increasing. Assume furthermore that there exists $C\in L^{\infty }_{\mathrm {loc}}((0,1))$, $\delta \in L^{\infty }_{\mathrm {loc}}({\mathbb {R}}_+)$ going to 0 at 0, and $f\in L^{\infty }_{\mathrm {loc}}({\mathbb {R}})$ such that for all $0<\sigma ,\tau <1$ and for all $x\in X$,

$$\begin{aligned} \Vert D F_{\tau }(x)-DF_{\sigma }(x)\Vert _x\le C(\sigma )\delta (|\sigma -\tau |)f(F_{\sigma }(x)). \end{aligned}$$

(1.2)

Finally, assume that for $\sigma >0$ the function $F_{\sigma }$ satisfies the Palais–Smale condition. Let ${\mathscr {A}}$ be an admissible family of min–max of X a nd denote

$$\begin{aligned} \beta (\sigma )=\beta (F_{\sigma },{\mathscr {A}})=\inf _{A\in {\mathscr {A}}}\sup F_{\sigma }(A). \end{aligned}$$

Then there exists a sequence ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}\subset (0,\infty )$ and ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that

$$\begin{aligned} F_{\sigma _k}(x_k)=\beta (\sigma _k),\quad DF_{\sigma _k}(x_k)=0. \end{aligned}$$

Furthermore, for all $k\in {\mathbb {N}}$, the critical point $x_k$ satisfies the following entropy condition

$$\begin{aligned} \partial _{\sigma }F_{\sigma _k}(x_k)\le \frac{1}{\sigma _k\log \left( \frac{1}{\sigma _k}\right) \log \log \left( \frac{1}{\sigma _k}\right) }. \end{aligned}$$

(1.3)

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 Sect. 2.2 for the definitions of index, nullity and of the different types of min–max families).

Theorem 1.1

Let $(X,\Vert \,\cdot \,\Vert _X)$ be a $C^2$ Finsler manifold modelled on a Banach space E, and $Y\hookrightarrow X$ be a $C^2$ Finsler–Hilbert manifold modelled on a Hilbert space H which we suppose locally Lipschitz embedded in X, and let $F,G\in C^2(X,{\mathbb {R}}_+)$ be two fixed functions. Define for all $\sigma >0$, $F_\sigma =F+\sigma ^2G\in C^2(X,{\mathbb {R}}_+)$ and suppose that the following conditions hold.

(1):

Palais–Smale condition For all $\sigma >0$, the function $F_{\sigma }:X\rightarrow {\mathbb {R}}_+$ satisfies the Palais–Smale condition at all positive level $c>0$.

(2):

Energy bound The following energy bound condition holds: for all $\sigma >0$ and for all ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that

$$\begin{aligned} \sup _{k\in {\mathbb {N}}}F_{\sigma }(x_k)<\infty , \end{aligned}$$

we have

$$\begin{aligned} \sup _{k\in {\mathbb {N}}}\Vert D G(x_k)\Vert <\infty , \end{aligned}$$

where $\Vert DG(x_k)\Vert =\sup _{y\in E, \Vert y\Vert _{E}\le 1}\langle DG(x_k),y\rangle$ where we identified $T_xX$ with E.

(3):

Fredholm property For all $\sigma >0$ and for all $x\in K(F_{\sigma })$, we have $x\in Y$, and the second derivative $D^2F_{\sigma }(x):T_xX\rightarrow T_x^{*}X$ restrict on the Hilbert space $T_xY$ such that the linear map $\nabla ^2F_{\sigma }(x)\in {\mathscr {L}}(T_xY)$ defined by

$$\begin{aligned} D^2F_{\sigma }(x)(v,v)=\langle \nabla ^2F_{\sigma }(x)v,v\rangle _{Y,x},\quad \text {for all}\;\, v\in T_xY, \end{aligned}$$

is a Fredholm operator, and the embedding $T_xY\hookrightarrow T_xX$ is dense for the Finsler norm $\Vert \,\cdot \,\Vert _{X,x}$.

Now, let ${\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, resp. $\overline{{\mathscr {A}}}$, resp. $\underline{{\mathscr {A}}}(\alpha _{*})$, resp. $\overline{{\mathscr {A}}}(\alpha ^{*})$, where the last two families depend respectively on a homology class $\alpha _{*}\in H_d(Y,B)$—where $B\subset Y$ is a fixed compact subset—and a cohomology class $\alpha ^{*}\in H^d (Y)$) be a d-dimensional admissible family of Y (resp. a d-dimension dual family to ${\mathscr {A}}$, resp. a d-dimensional co-dual family to ${\mathscr {A}}^{*}$, resp. a d-dimensional homological family, resp. a d-dimensional co-homological family) with boundary ${\left\{ C_i\right\} }_{i\in I}\subset Y$. Define for all $\sigma >0$

$$\begin{aligned} \begin{aligned}&\beta (\sigma )=\inf _{A\in {\mathscr {A}}}\sup F_\sigma (A)<\infty ,\quad&\beta ^{*}(\sigma )=\inf _{A\in {\mathscr {A}}^{*}}\sup F_{\sigma }(A), \quad&{\widetilde{\beta }}(\sigma )=\inf _{A\in \widetilde{{\mathscr {A}}}}\sup F_{\sigma }(A)\\&{\overline{\beta }}(\sigma )=\inf _{A\in \underline{{\mathscr {A}}}(\alpha _{*})}\sup F_{\sigma }(A),\quad&{\underline{\beta }}(\sigma )=\inf _{A\in \overline{{\mathscr {A}}}(\alpha ^{*})}\sup F_{\sigma }(A). \end{aligned} \end{aligned}$$

Assuming that the min–max value is non-trivial, i.e.

(4):

Non-trivialilty $\beta _0=\inf _{A\in {\mathscr {A}}}\sup F(A)>\sup _{i\in I} \sup F(C_i)={\widehat{\beta }}_0$,

there exists a sequence ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}\subset (0,\infty )$ such that $\sigma _k{\underset{k\rightarrow \infty }{\longrightarrow }}0$, and for all $k\in {\mathbb {N}}$, there exists a critical point $x_k\in K(F_{\sigma _k})\cap {\mathscr {E}}(\sigma _k)$ (resp. $x_k^{*},{\widetilde{x}}_k,{\underline{x}}_k,{\overline{x}}_{k}\in K(F_{\sigma _k})\cap {\mathscr {E}}(\sigma _k)$) of $F_{\sigma _k}$ (where we recall that $x_k\in Y$ by the condition (3)) satisfying the entropy condition (1.3) (see Definition 4.4) and such that respectively

$$\begin{aligned} \left\{ \begin{array}{llll} F_{\sigma _k}(x_k)=\beta (\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}(x_k)\le d\\ F_{\sigma _k}(x_k^{*})=\beta ^{*}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}(x_{k})\ge d \\ F_{\sigma _k}({\widetilde{x}}_k)={\widetilde{\beta }}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}({\widetilde{x}}_k)\le d\le \mathrm {Ind}_{F_{\sigma _k}}({\widetilde{x}}_k)+\mathrm {Null}_{F_{\sigma _k}}({\widetilde{x}}_k) \\ F_{\sigma _k}({\overline{x}}_k)={\overline{\beta }}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}({\overline{x}}_k)\le d\le \mathrm {Ind}_{F_{\sigma _k}}({\overline{x}}_k)+\mathrm {Null}_{F_{\sigma _k}}({\overline{x}}_k) \\ F_{\sigma _k}({\underline{x}}_k)={\underline{\beta }}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}({\underline{x}}_k)\le d\le \mathrm {Ind}_{F_{\sigma _k}}({\underline{x}}_k)+\mathrm {Null}_{F_{\sigma _k}}({\underline{x}}_k). \end{array}\right. \end{aligned}$$

Remark 1.2

The previous theorem is stated for a family $F_{\sigma }=F+\sigma ^2G$, but it would hold more generally under the hypotheses of the previous Theorem $(*)$ of families $F_{\sigma }$, $C^1$ and increasing with respect to $\sigma$. Notice that the Energy Bound is nothing else that the bound (1.2) in this particular case.

Remarks 1.3

(On the optimality of the hypotheses of Theorem 1.1) Firstly, the Palais–Smale condition might be weakened to the Palais–Smale condition along certain near-optimal sequence (see [8]). However, the sequence ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}$ given by the theorem cannot be made explicit, as is depends on differentiability property of $\sigma \mapsto \beta (\sigma )$ (actually, of certain approximations of this function), a function which is a priori impossible to determine explicitly for all $\sigma > 0$ (determining $\beta (0)$ 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 [7, 8, 12, 34, 38]. The Fredholm property is probably necessary as all existing methods rely on perturbation methods using the Sard–Smale theorem [32], for which the Fredholm hypothesis is necessary, thanks to the counter-example of Kupka [10]. Furthermore, we have to make the hypothesis that $T_xY$ be dense in $T_xX$ for a critical point $x\in K(F_{\sigma })$ as it shows that the index does not change for the restriction $\nabla ^2F_{\sigma }(x)\in {\mathscr {L}}(T_xY)$.

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 [34] 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.

Remark 1.4

Theorem 1.1 would still be valid if $(X,\Vert \,\cdot \,\Vert _X)$ depended on $\sigma$.

1.1 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 [20]. The only case to check is the one of homotopy classes of mappings. Let $M^d$ be a smooth manifold and let c a regular homotopy class of immersions of $M^d$ into X, or an isotopy class of embeddings of $M^d$ into X. Then

$$\begin{aligned} {\mathscr {A}}(c)={\left\{ f(M^d): f\in c\right\} } \end{aligned}$$

is ambient isotopy invariant so is an admissible family of dimension d, i.e. one may freely have additional constraints in the definition of the admissible families as long as they are stable under homeomorphisms isotopic to the identity (preserving the boundary conditions, if any). In particular, if $\Sigma ^k, N^n$ are two smooth manifolds, $\mathrm {Imm}(\Sigma ^k,N^n)$ is the set of smooth immersions from $\Sigma ^k$ to $N^n$, and $d\in {\mathbb {N}}$ is such that

$$\begin{aligned} \pi _d\left( \mathrm {Imm}(\Sigma ^k,N^n)\right) \ne {\left\{ 0\right\} }, \end{aligned}$$

where $\pi _d$ designs the d-th regular homotopy group, then for all $c\in \pi _d\left( \mathrm {Imm}(\Sigma ^k,N^n)\right)$ with $c\ne 0$, and for all $l\in {\mathbb {N}}$ and $1\le p<\infty$ such that $lp>k$, as the following Sobolev space of immersion is a smooth Banach manifold [24]

$$\begin{aligned} \mathrm {Imm}_{l,p}(\Sigma ^k,N^n)=W^{l,p}(\Sigma ^k,N^n)\cap {\left\{ \vec {\Phi }:d\vec {\Phi }(p)\wedge d\vec {\Phi }(p)\ne 0\;\,\text {for all}\;\, p\in \Sigma ^k\right\} }, \end{aligned}$$

we deduce that

$$\begin{aligned} {\mathscr {A}}(c)={\mathscr {P}}(\mathrm {Imm}_{l,p}(\Sigma ^k,N^n))\cap {\left\{ \vec {\Phi }(S^d): \vec {\Phi }\in C^0(S^d,\mathrm {Imm}_{l,p}(\Sigma ^k,N^n)), [\vec {\Phi }]=c\right\} } \end{aligned}$$

is a d-dimensional min–max family of $\mathrm {Imm}_{l,p}(\Sigma ^k,N^n)$.

1.2 Applications

Sacks-Uhlenbeck $\alpha$-energies [31]. Let $\Sigma$ be a closed Riemann surfaces and let $(M^n,h)$ be a closed Riemannian manifold which we suppose isometrically embedded in some Euclidean space ${\mathbb {R}}^N$, and define for all $\sigma \ge 0$ the family of Banach spaces

$$\begin{aligned} X_{\sigma }&=W^{1,2(1+\sigma )}(\Sigma ,M^n)=W^{1,1+\sigma }(\Sigma ,{\mathbb {R}}^N)\cap {\left\{ \vec {u}: \vec {u}(p)\in M^n\;\,\text {for a.e.}\;\, p\in \Sigma \right\} }.\\ Y&=W^{2,2}(\Sigma ,M^n). \end{aligned}$$

One can check that also $X_{\sigma }$ depends on $\sigma$, as Y is independent of $\sigma$, the proof of Theorem 1.1 is still valid. The function $F_{\sigma }:X_{\sigma }\rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} F_{\sigma }(\vec {u})=\frac{1}{2}\int _{\Sigma }\left( \left( 1+|d\vec {u}|^2\right) ^{1+\sigma }-1\right) d\mathrm {vol}_{g_0} \end{aligned}$$

where $g_0$ is some fixed smooth metric on $\Sigma$.

The significance of the restriction on the Hilbert space Y is given by the following regularity result (see [17]).

Theorem

If $0<\sigma <1/2$, any critical point $\vec {u}\in X_{\sigma }$ of $F_{\sigma }$ is smooth. Furthermore, for all $0<\sigma <1/2$, and all critical point $\vec {u}\in K(F_{\sigma })$, the restriction $D^2F_{\sigma }(\vec {u}):T_{\vec {u}}Y\rightarrow T_{\vec {u}}^{*}Y$ is a Fredholm operator.

In particular, such critical point $u\in X_{\sigma }$ 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 Uhlenbeck [37].

In order to introduce the next two categories, we introduce some additional definitions. Let $\Sigma$ be a closed Riemann surfaces of genus $\gamma$, and $\mathrm {Diff}_+^{*}(\Sigma )$ be the topological group of positive $W^{3,2}$-diffeomorphism (we adopt the standard notations of e.g. [3] for Sobolev functions) with either 3 distinct marked points if $\gamma =0$, or 1 marked point for $\gamma =1$ and no mark points for higher genera. Furthermore, if $(M^n,h)$ is a fixed Riemannian manifold, we denote by $\mathrm {Imm}(\Sigma ,M^n)$ the Banach manifold of $W^{k,p}$-immersions (for $kp>2$) by

$$\begin{aligned} \mathrm {Imm}_{{k,p}}(\Sigma ,M^n)=W^{k,p}(\Sigma ,M^n)\cap {\left\{ \vec {\Phi }:d\vec {\Phi }(p)\wedge d\vec {\Phi }(p)\ne 0\;\, \text {for all}\;\, p\in \Sigma \right\} } \end{aligned}$$

It was recently proved by Rivière [27] that the quotient spaces

$$\begin{aligned}&X=\widetilde{\mathrm {Imm}}_{2,4}(\Sigma ,M^n)=\mathrm {Imm}_{2,4}(\Sigma ,M^n)/\mathrm {Diff}_+^{*}(\Sigma )\\&Y=\widetilde{\mathrm {Imm}}_{3,2}(\Sigma ,M^n)=\mathrm {Imm}_{3,2}(\Sigma ,M^n)/\mathrm {Diff}_+^{*}(\Sigma ) \end{aligned}$$

are (respectively) separated smooth Banach and Hilbert manifolds, and this is really a crucial fact, as by the invariance under the diffeomorphism group on $\Sigma$, 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 [21, 22, 28, 29]. Here, the Finsler manifolds are

$$\begin{aligned} X=\widetilde{\mathrm {Imm}}_{2,4}(\Sigma ,M^n),\quad Y=\widetilde{\mathrm {Imm}}_{3,2}(\Sigma ,M^n) \end{aligned}$$

and the functions

$$\begin{aligned} F(\vec {\Phi })=\mathrm {Area}(\vec {\Phi }(\Sigma ))=\int _{\Sigma }d\mathrm {vol}_g,\quad G(\vec {\Phi })=\int _{\Sigma }\left( 1+|\vec {{\mathbb {I}}}_g|^2\right) ^2d\mathrm {vol}_g \end{aligned}$$

where $g=\vec {\Phi }^{*}h$ is the pull-back of the metric h on $M^n$ by the immersion $\vec {\Phi }$, and $\vec {{\mathbb {I}}}_g$ is the second fundamental form of the immersion $\vec {\Phi }:\Sigma \rightarrow M^n$. However, we see that the subtlety here is that $F_{\sigma }$ satisfies the Palais–Smale condition only on X, but not on Y. However, as for all critical point $\vec {\Phi }\in X$, we actually have $\vec {\Phi }\in C^{\infty }(\Sigma ,M^n)$, then we have in particular $\vec {\Phi }\in Y$, and one can directly verify that $D^2F_{\sigma }(\vec {\Phi })$ is Fredholm on the Hilbert space $T_{\vec {\Phi }}Y$ (see [27]). 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 Pigati and Rivière [21] with the previous result of Rivière [27], one can obtain the lower semi-continuity of the index.

Willmore surfaces [14, 16, 26].

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

$$\begin{aligned} X=\widetilde{\mathrm {Imm}}_{2,4}(S^2,{\mathbb {R}}^n),\quad Y=\widetilde{\mathrm {Imm}}_{3,2}(S^2,{\mathbb {R}}^n) \end{aligned}$$

and

$$\begin{aligned} F(\vec {\Phi })&=\mathrm {W}(\vec {\Phi })=\int _{S^2}|\vec {H}_g|^2d\mathrm {vol}_g,\\ F_{\sigma }(\vec {\Phi })&=F(\vec {\Phi })+\sigma ^2\int _{S^2}\left( 1+|\vec {H}_g|^2\right) ^2d\mathrm {vol}_g+\frac{1}{\log \left( \frac{1}{\sigma }\right) }{\mathscr {O}}(\vec {\Phi }) \end{aligned}$$

where $\vec {H}_g$ is the mean-curvature of the immersion $\vec {\Phi }:S^2\rightarrow {\mathbb {R}}^n$, and ${\mathscr {O}}(\vec {\Phi })$ is the Onofri energy, defined by

$$\begin{aligned} {\mathscr {O}}(\vec {\Phi })=\frac{1}{2}\int _{S^2}|d\alpha |_g^2d\mathrm {vol}_g+4\pi \int _{S^2}\alpha e^{-2\alpha }d\mathrm {vol}_g-2\pi \log \left( \int _{S^2}d\mathrm {vol}_g\right) \ge 0. \end{aligned}$$

where $\alpha :S^2\rightarrow {\mathbb {R}}$ is the function given by the Uniformisation Theorem such that $g=e^{2\alpha }g_0$, 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 hypotheses 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].

Min–max hierarchies for minimal surfaces [27, 29].

We observe that the previously considered admissible families do not need to 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 Rivière [27, 29]. We first introduce some terminology (see [6] chapter 4, [23] chapter 2, [1] section 3).

Let $\Sigma$ 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 ${\mathscr {G}}_2(TN^n)$ be the Grassmannian bundle of oriented 2-planes in $TN^n$. We denote by ${\mathscr {V}}_2(N^n)$ the space of 2-dimensional varifolds on $N^n$, that is the space of Radon measure on ${\mathscr {G}}_2(TN^n)$ endowed with the weak-$*$ topology. Furthermore, we denote by ${\mathscr {Z}}_2(N^n,G)$ the space of rectifiable 2-cycles in $N^n$ with G-coefficients (see [6], 4.1.24, 4.2.26, 4.4.1), where $G={\mathbb {Z}}$ or $G={\mathbb {Z}}_2$ (or more generally, G is an admissible in Almgren’s sense [2]). It is known that every current $T\in {\mathscr {Z}}_k(N^n,G)$ induces a varifold $|T|\in {\mathscr {V}}_2(N^n)$, and we denote by ${\mathcal {F}}$ the flat norm on ${\mathscr {Z}}_2(N^n,G)$ and by $d_{{\mathscr {V}}}$ the varifold distance, defined for all $V,W\in {\mathscr {V}}_2(N^n)$ by

$$\begin{aligned} d_{{\mathscr {V}}}(V,W)=\sup {\left\{ V(f)-W(f):\, f\in C^0_c({\mathscr {G}}_2(TN^n)),\;\, \Vert f\Vert _{\mathrm {L}^{\infty }}\le 1,\;\, \mathrm {Lip}(f)\le 1\right\} }. \end{aligned}$$

Furthermore, if $\vec {\Phi }\in \mathrm {Imm}_{3,2}(\Sigma ,N^n)$ is a $W^{3,2}$ immersion as defined in Sect. 1.2, then obviously the push-forward $\vec {\Phi }_{*}[\Sigma ]$ of the current of integration $[\Sigma ]$ on the closed Riemann surface $\Sigma$ is an element of ${\mathscr {Z}}_2(N^n,{\mathbb {Z}})$, and furthermore, the induced varifold is denoted by $V_{\vec {\Phi }}=|\vec {\Phi }_{*}[\Sigma ]|\in {\mathscr {V}}_2(N^n)$. We have explicitly for all $f\in C^0_{c}({\mathscr {G}}_2(TN^n))$

$$\begin{aligned} V_{\vec {\Phi }}(f)=\int _{\Sigma }f\left( \Phi (p), \vec {\Phi }_{*}T_p\Sigma \right) d\mathrm {vol}_g(p). \end{aligned}$$

We introduce the following distance on ${\mathscr {V}}_2(N^n)\cap {\left\{ |T|: T\in {\mathbb {Z}}_2(N^n,G)\right\} }$: for all $V,W\in {\mathscr {V}}_2(N^n)$ such that $V=|S|$ and $W=|T|$ for some $S,T\in {\mathscr {Z}}_2(N^n,G)$,

$$\begin{aligned} {\mathbf {F}}(S,T)=d_{{\mathscr {V}}}(|S|,|T|)+{\mathcal {F}}(S,T). \end{aligned}$$

Finally, if for all $g\in {\mathbb {N}}$, $\Sigma _g$ is a fixed closed oriented surface of genus g, we denote by $\mathrm {Imm}_{3,2}^0(\Sigma _g,N^n)$ the connected component (for regular homotopy) of the immersions regularly homotopic to an embedding $\Sigma _g\hookrightarrow N^n$, on we denote by $\mathrm {Imm}^{\le g_0}(N^n)$ the disjoint union of Finsler–Hilbert manifolds

$$\begin{aligned} \mathrm {Imm}^{\le g_0}(N^n)=\bigsqcup _{g=0}^{g_0}\mathrm {Imm}_{3,2}^0(\Sigma _g,N^n), \end{aligned}$$

We introduce for all $0\le \sigma \le 1$ the function $A_{\sigma }:\mathrm {Imm}^{\le g_0}(N^n)\rightarrow {\mathbb {R}}$ defined for all $\vec {\Phi }\in \mathrm {Imm}^{\le g_0}(N^n)$ by

$$\begin{aligned} A_{\sigma }(\vec {\Phi })=\mathrm {Area}(\Phi (\Sigma ))+\sigma ^2\int _{\Sigma }\left( 1+|\vec {{\mathbb {I}}}_{\vec {\Phi }}|^2\right) ^2d\mathrm {vol}_g\end{aligned}$$

if $\vec {\Phi }$ is defined from a closed surface $\Sigma$, and $\vec {{\mathbb {I}}}_{\vec {\Phi }}$ is its second fundamental form. That $A_{\sigma }$ satisfies all the hypotheses of Theorem 1.1 is verified in [28].

Corollary 1.5

Let $N^n$ be a closed Riemannian manifold, I be a non-empty set and let ${\left\{ M_i^d\right\} }_{i\in I}$ a family of d-dimensional cellular-complexes, for all $i\in I$, let $h_i:\partial M_{i}^d\rightarrow \mathrm {Imm}^{\le g_0}(N^n)$ by a ${\mathbf {F}}$-Lipschitzian map, and define

$$\begin{aligned} {\mathscr {A}}=\mathrm {Imm}^{\le g_0}(N^n)\cap {\left\{ \vec {\Phi }(Y): \vec {\Phi }\in \mathrm {Lip}_{{\mathbf {F}}}(M_i^d,\mathrm {Imm}^{\le g_0}(N^n))\;\, \text {for some}\;\, i\in I\right\} }, \end{aligned}$$

and define for all $0\le \sigma \le 1$

$$\begin{aligned} \beta (\sigma )=\beta (A_{\sigma },{\mathscr {A}})=\inf _{A\in {\mathscr {A}}}\sup A_{\sigma }(A)<\infty . \end{aligned}$$

Assuming that ${\mathscr {A}}$ is non-trivial as in Theorem 1.1, there exists a sequence ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}\subset (0,\infty )$ such that $\sigma _k\rightarrow 0$ and and for all $k\in {\mathbb {N}}$, there exists a critical point $x_k\in K(A_{\sigma _k})\cap {\mathscr {E}}(\sigma _k)$ such that

$$\begin{aligned} A_{\sigma _k}(x_k)&=\beta (\sigma _k),\\ \sigma _k^2\int _{\Sigma _{\vec {\Phi }_k}}\left( 1+|\vec {{\mathbb {I}}}_{\vec {\Phi }_k}|^2\right) ^2d\mathrm {vol}_{g_{\vec {\Phi }_k}}&\le \frac{1}{\log \left( \frac{1}{\sigma _k}\right) \log \log \left( \frac{1}{\sigma _k}\right) }, \quad \mathrm {Ind}_{A_{\sigma _k}}(\vec {\Phi }_k)\le d. \end{aligned}$$

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 Lipschitzian in the strong topology on $W^{3,2}$ immersions, so the proof is virtually unchanged. $\square$

1.3 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 $\sigma >0$, the approximation $F_{\sigma }$ is non-degenerate and in the second step that $\nabla ^2F(x):T_xX\rightarrow T_xX$ is a Fredholm map at every critical point $x\in K(F_{\sigma })$. 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.

2 Technical lemmas

2.1 Preliminary definitions

Definition 2.1

Let $\pi :{\mathscr {E}}\rightarrow X$ be a Banach space bundle over a Banach manifold X and let $\Vert \,\cdot \,\Vert :{\mathscr {E}}\rightarrow {\mathbb {R}}_+$ be a continuous function such that for all $x\in X$ the restriction $\Vert \,\cdot \,\Vert _{x}$ is a norm on the fibre ${\mathscr {E}}_x=\pi ^{-1}({\left\{ x\right\} })$. For all $x_0\in X$, for all trivialisation $\varphi _{x_0}:\pi ^{-1}(U_{x_0})\rightarrow U_{x_0}\times {\mathscr {E}}_{x_0}$ (where U is some open neighbourhood of $x_0$) and for all $x\in U$, we get an isomorphism $L_x:{\mathscr {E}}_{x}\rightarrow {\mathscr {E}}_{x_0}$ so $\Vert \,\cdot \,\Vert _x$ induces a norm on ${\mathscr {E}}_{x_0}$ by

$$\begin{aligned} \Vert v\Vert _{x}=\Vert L_x^{-1}(v)\Vert _{x}\quad (\text {for all}\;\,v\in {\mathscr {E}}_{x_0}). \end{aligned}$$

We say that $\Vert \,\cdot \,\Vert :{\mathscr {E}}\rightarrow {\mathbb {R}}$ is a Finsler structure on ${\mathscr {E}}$ if for all $x_0\in X$ and all such trivialisation $(U_{x_0},\varphi _{x_0})$, there exists a constant $C=C_{x_0}\ge 1$ such that for all $x\in U_{x_0}$,

$$\begin{aligned} \frac{1}{C}\Vert \,\cdot \,\Vert _x\le \Vert \,\cdot \,\Vert _{x_0}\le C\Vert \,\cdot \,\Vert _x. \end{aligned}$$

A Finsler manifold is a regular (in the topological sense) $C^1$ Banach manifold X equipped with a Finsler structure on the tangent space TX. A Finsler–Hilbert manifold or (infinite-dimensional) Riemannian manifold is a Finsler manifold modelled on a Hilbert space.

Theorem 2.2

(Palais [20]) Let $(X,\Vert \,\cdot \,\Vert )$ be a Finsler manifold, and $d:X\times X\rightarrow {\mathbb {R}}_+\cup {\left\{ \infty \right\} }$ be such that for all $x,y\in X$

$$\begin{aligned} d(x,y)=\inf {\left\{ \int _{0}^{1}\Vert \gamma '(t)\Vert _{\gamma (t)}dt: \gamma \in C^0([0,1],X), \gamma (0)=x,\;\, \gamma (1)=y\right\} }. \end{aligned}$$

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 distances, usually denoted by d, and we will denote for all $A\subset X$ and $\delta >0$

$$\begin{aligned}&N_{\delta }(A)=X\cap {\left\{ x: d(x,A)\le \delta \right\} }\\&U_{\delta }(A)=X\cap {\left\{ x: d(x,A)<\delta \right\} }. \end{aligned}$$

Theorem 2.3

(Palais [20]) Let $(X,\Vert \,\cdot \,\Vert )$ be a Finsler manifold modelled on some Banach space E, let $U\subset X$ be an open subset, $\varphi :U\rightarrow E$ be a chart and $x_0\in U$. We define for all $r>0$

$$\begin{aligned}&B(x_0,r)=U\cap {\left\{ x:\Vert \varphi (x)-\varphi (x_0)\Vert \le r\right\} }\\&U(x_0,r)=U\cap {\left\{ x: \Vert \varphi (x)-\varphi (x_0)\Vert <r\right\} }\\&S(x_0,r)=U\cap {\left\{ x: \Vert \varphi (x)-\varphi (x_0)\Vert =r\right\} }. \end{aligned}$$

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 $(X,\Vert \,\cdot \,\Vert )$ be a Finsler manifold and $K\subset X$ be a compact subset. Then for $r>0$ small enough, $N_{\delta }(K)$ is closed, and $U_{\delta }(K)$ is its interior relative to X.

Definition 2.5

Let E, F be two Banach spaces. We say that a linear map $T\in {\mathscr {L}}(E,F)$ is a Fredholm operator if $\mathrm {Im}\,(T)\subset F$ is closed, and $\mathrm {Ker}(T)\subset E$ and $\mathrm {Coker}(T)=F/\mathrm {Im}(T)$ are finite-dimensional. Then the index $\mathrm {Ind}(T)\in {\mathbb {Z}}$ is defined by

$$\begin{aligned} \mathrm {Ind}(T)=\dim \mathrm {Ker}(T)-\mathrm {dim}(\mathrm {Coker}(T)). \end{aligned}$$

Definition 2.6

Let X, Y be two Banach manifolds and $F:X\rightarrow Y$ be a $C^1$ map. We say that F is a Fredholm map at x if $DF(x):T_xX\rightarrow T_{F(x)}Y$ is a Fredholm operator and we define the index of F at x, still denoted by $\mathrm {Ind}_x(F)$, by

$$\begin{aligned} \mathrm {Ind}_x(F)=\mathrm {Ind}(DF(x)). \end{aligned}$$

As the map $x\mapsto \mathrm {Ind}_x(F)\in {\mathbb {Z}}$ is continuous, we deduce that it is constant on each connected component of X, and we will denote it by $\mathrm {Ind}(F)$ if F is defined on a connected domain.

In the applications we have in mind, we cannot assume that the manifold X is connected, so we will have to keep in mind this technical point.

If $F:X\rightarrow Y$ is a $C^1$ map between Banach manifolds, we say that $x\in X$ is a regular point if $DF(x):X\rightarrow Y$ is surjective. The complement of the regular points are called the singular points, the image under F of the singular points are the critical values and their complement the regular values.

Now we recall the celebrated Sard’s theorem of Smale, which proceeds by reducing the infinite dimensional version to the finite dimensional Sard’s theorem.

Theorem 2.7

(Smale [33]) Let X, Y be two Banach manifolds and let $U\subset X$ be an open connected subset and $F:U\rightarrow Y$ be a $C^q$ Fredholm map, where

$$\begin{aligned} q\ge \max {\left\{ \mathrm {Ind}(F),0\right\} }+1. \end{aligned}$$

Then the regular values of F are almost all Y, i.e. the set of critical value is a set of first Baire category (or meagre).

2.2 Morse index and admissible families of min–max

Let X a $C^2$ Banach manifold, and suppose that $F:X\rightarrow {\mathbb {R}}$ is a function which admits second order Gâteaux derivatives in X, i.e. for all $C^2$ path $\gamma :(-\varepsilon ,\varepsilon )\rightarrow X$ the function $t\mapsto F(\gamma (t))$ is a $C^2$ function. Then a critical point $x\in X$ of F is an element such that for all $C^2$ path $\gamma :(-\varepsilon ,\varepsilon )\rightarrow X$ such that $\gamma (0)=x$, we have

$$\begin{aligned} \frac{d}{dt}F(\gamma (t))_{|t=0}=0. \end{aligned}$$

If x is a critical point, we define the second derivative quadratic form $Q_x=D^2F(x):T_xX\rightarrow (T_xX)^{*}$ by

$$\begin{aligned} Q_x(v)(v)=D^2F(x)(v,v)=\frac{d^2}{dt^2}F(\gamma (t))_{|t=0} \end{aligned}$$

for all $v\in T_xX$ and path $\gamma :(-\varepsilon ,\varepsilon )\rightarrow X$ such that $\gamma (0)=x$ and $\gamma '(0)=v$.

Then $Q_x$ is a well-defined continuous map on $T_xX$, and the index $\mathrm {Ind}_{F}(x)$ of x with respect to F, is defined by

$$\begin{aligned} \mathrm {Ind}_{F}(x)=\sup {\left\{ \dim V:V\subset T_xX\;\text {is a sub vector-space such that}\; Q_x(v)(v)<0\;\text {for all}\; v\in V\ {0}\right\} } \end{aligned}$$

To define the nullity, we need to assume that $F:\rightarrow {\mathbb {R}}$ is a $C^2$ Fréchet differentiable map and recalling that $Q_x=D^2F(x):T_xX\rightarrow (T_xX)^{*}$, we define

$$\begin{aligned} \mathrm {Null}_F(x)=\sup {\left\{ \dim W:W\subset T_xX\;\text {is a sub vector-space such that}\; Q_x(w)=0\;\text {for all}\;w\in W\right\} }. \end{aligned}$$

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

$$\begin{aligned} D^2F(x)(v,v)=\langle L v,v\rangle _x \end{aligned}$$

for some self-adjoint linear operator $L:T_xX\rightarrow T_xX$. 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. $\mathrm {Null}_F(x)=\dim \mathrm {Ker} (L)$).

Important remark 2.8

In particular, if $Y\subset X$ is a Lipschitz embedded Hilbert manifold, and $x\in Y$ is a critical point of F, then the square gradient $\nabla ^2F(x):T_{x}X\rightarrow T_{x}X$ restricts continuously to the Hilbert space $T_{x}Y$ and the definition of the index is unchanged, provided that $T_xY\subset T_{x}X$ is dense, a condition easily verified in the cases of interest to us (it is stated explicitly in the hypotheses of 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 ${\mathscr {A}}\subset {\mathscr {P}}(X){\setminus }{\left\{ \varnothing \right\} }$ is an admissible min–max family of dimension $d\in {\mathbb {N}}$ with boundary $(B^{d-1},h)$ (possibly empty) for X if

  • (A1) For all $A\in {{\mathscr {A}}}$, A is compact in X,

  • (A2) There exists a d-dimensional compact Lipschitz manifold $M^d$ with boundary $B^{d-1}$, (possibly empty) and a continuous map $h:B^{d-1}\rightarrow X$ such that for all $A\in {\mathscr {A}}$, there exists a continuous map $f:M^d\rightarrow X$ that $A=f(M^d)$, and $f=h$ on $B^{d-1}$.

  • (A3) For every homeomorphism $\varphi$ of X isotopic to the identity map such that $\varphi |_{B^{d-1}}=\mathrm {Id}|_{h(B^{d-1})}$, and for all $A\in {{\mathscr {A}}}$, we have $\varphi (A)\in {\mathscr {A}}$.

  More generally, one can relax the notions of uniqueness of the compact manifold $M^d$ as follows. Let I a set of indices and a family ${\left\{ M_i^d\right\} }_{i\in I}$ of compact Lipschitz manifold with boundary $(B^{d-1}_i,h_i)$. Then we define

  $$\begin{aligned} {\mathscr {A}}&={\mathscr {P}}(X)\cap \big \{A: \text {there exists}\;i\in I\;\text {and}\; f\in C^0(M_i^d,X)\; \text {such that}\; A=f(M_i^d)\\&\quad \text {and}\; f_{B_i^{d-1}}=h_i\big \} \end{aligned}$$

  Clearly, these classes are stable under homeomorphisms of X isotopic to the identity preserving the boundary h(B) (resp. $h(B_i)$ for all $i\in I$).

• (2) Dual admissible family. In a dual fashion, let I be a (non-empty) set of indices and let ${\left\{ C_i\right\} }_{i\in I}\subset X$ be a collection of subsets such that for all $i\in I$, there exists a non-empty set $J_i$ and a family of continuous functions $\{h_i^j\}_{j\in J_i}\in C^0(C_i,{\mathbb {R}}^d)$. Then we define ${\mathscr {A}}^{*}={\mathscr {A}}(I,{\left\{ J_i\right\} }_{i\in I}, \{h_i^j\}_{i\in I, j\in J_i})$ by

  $$\begin{aligned} {\mathscr {A}}^{*}&={\mathscr {P}}(X)\cap \{A: \text { there exists } i\in I\;\, \text {such that for all } h\in C^0(X,{\mathbb {R}}^d)\;\\ {}&\qquad \text {such that } h_{|C_i}=h_{i}^j\; \text {for some } j\in I \quad \text {one has}\; 0\in h(A) \}. \end{aligned}$$

  If the functions $h_i:B^{d-1}_i\rightarrow X$ are implicit, then we say by abuse of notation that ${\left\{ C_i\right\} }_{i\in I}={\left\{ h(B_i^{d-1})\right\} }_{i\in I}$ is the boundary of ${\mathscr {A}}$ (this permits to give a uniform definition of boundary for each admissible family).

• (3) Co-dual admissible family. Finally, given a d-dimension dual admissible family ${\mathscr {A}}^{*}$, a d-dimensional co-dual admissible family is defined by

  $$\begin{aligned} \widetilde{{\mathscr {A}}}={\mathscr {A}}^{*}\cap {\left\{ A:\dim _{{\mathscr {H}}}(A)<d+1\right\} }, \end{aligned}$$

  where $\dim _{{\mathscr {H}}}$ 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 Lipschitzian homeomorphism of X isotopic to the identity (this is not restrictive, as the only homeomorphisms of interest are pseudo-gradient flow of $C^2$ functions, which are indeed locally Lipschitzian).

Finally, we define the following boundary values of admissible families ${\mathscr {A}}$, ${\mathscr {A}}^{*}$ and $\widetilde{{\mathscr {A}}}$ with boundary ${\left\{ C_i\right\} }_{i\in I}$ by

$$\begin{aligned} {\widehat{\beta }}(F,{\mathscr {A}})&=\sup _{i\in I}\,\sup F(h_i(B_i^{d-1})),\quad (\text {where}\;\, C_i=h(B^{d-1}_i)\;\, \text {for all}\;\, i\in I)\\ {\widehat{\beta }}(F,{\mathscr {A}}^{*})&={\widehat{\beta }}(F,\widetilde{{\mathscr {A}}})=\sup _{i\in I}\,\sup F(C_i). \end{aligned}$$

Remark 2.10

The definition of the third family in [12] is the more restrictive

$$\begin{aligned} \widetilde{{\mathscr {A}}}={\mathscr {A}}^{*}\cap {\left\{ A: {\mathscr {H}}^d(A)<\infty \right\} } \end{aligned}$$

but as we shall see, our definition will still permit to obtain the suitable two-sided index bounds.

Remark 2.11

In the definition of the first family of min–max, the hypotheses on $M^d$ (or equivalently on ${\left\{ M_i^d\right\} }_{i\in I}$) can be considerably weakened, as the main Theorem 1.1 would still hold if $M^d$ were merely a metric space of Hausdorff dimension (with respect to the metric) at most d admitting Lipschitzian partitions of unity. Furthermore, the family of boundaries ${\left\{ B_i^{d-1}\right\} }_{i\in I}$ 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. It is particularly important in the example of Section 1.2, 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 ${\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, resp. $\widetilde{{\mathscr {A}}}$) be a d-dimensional admissible (resp. dual, resp. co-dual) min–max family with boundary ${\left\{ C_i\right\} }_{i\in I}$. We say that ${\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, resp. $\widetilde{{\mathscr {A}}}$) is non-trivial with respect to a continuous map $F:X\rightarrow {\mathbb {R}}$ if

$$\begin{aligned} \beta (F,{\mathscr {A}})=\inf _{A\in {\mathscr {A}}}\sup F(A)>\sup _{i\in {I}} \sup F(h_i(B_i^{d-1}))={\widehat{\beta }}(F,{\mathscr {A}}). \end{aligned}$$

(2.1)

Whenever this does not yield confusion, we shall write more simply $\beta ({\mathscr {A}})$ and ${\widehat{\beta }}({\mathscr {A}})$.

Remark 2.13

The condition $\mathrm {(A2)}$ can be relaxed in the sense that the applications $f:M^d\rightarrow X$ 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 1.2 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 $d\in {\mathbb {N}}$ be a fixed integer.

• (4) Homological family. Let ${\alpha }_{*}\in H_d(X,B,R){\setminus } {\left\{ 0\right\} }$ be a non-trivial d-dimensional relative (singular) homology class of X with respect to B with R coefficients. We say that $\underline{{\mathscr {A}}}=\underline{{\mathscr {A}}}(\alpha _{*})$ is a d-dimensional homological family with respect to $\alpha _{*}\in H_d(X,B,R)$ and boundary B if

  $$\begin{aligned} \underline{{\mathscr {A}}}({\alpha }_{*})={\mathscr {P}}(X)\cap {\left\{ A: A\;\text {compact}, B\subset A\; \text {and}\; \alpha \in \mathrm {Im}(\iota ^{A}_{*})\right\} }, \end{aligned}$$

  where for all $A \supset B$, the application $\iota _{*}^A: H_d(A,B,R)\rightarrow H_d(X,A,R)$ is the induced map in homology from the injection $\iota ^{A}:A\rightarrow X$.

• (5) Cohomological family. Let $\alpha ^{*}\in H^{d}(X,G){\setminus }{\left\{ 0\right\} }$ be a non-trivial d-dimensional (singular) cohomology class of X with G coefficients. We say that $\overline{{\mathscr {A}}}=\overline{{\mathscr {A}}}(\alpha ^{*})$ is a d-dimensional cohomological family with respect to $\alpha ^{*}\in H^d(X,G)$ if

  $$\begin{aligned} \overline{{\mathscr {A}}}(\alpha ^{*})={\mathscr {P}}(X)\cap {\left\{ A: A \text { compact and}\; \alpha ^{*}\notin \mathrm {Ker}(\iota ^{*}_{A})\right\} }, \end{aligned}$$

  where for all $A\subset X$, the application $\iota ^{*}_{ A}:H^d(X,G)\rightarrow H^d(A,G)$ is the induced map in cohomology from the injection $\iota _{A}:A\rightarrow X$. In other word, the non-zero class $\alpha ^{*}$ is not annihilated by the restriction map in cohomology $\iota ^{*}_{ A}:H^d(X,G)\rightarrow H^d(A,G)$.

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 $\mathrm {Id}_{X}:X\rightarrow X$. One can check that no restrictions is necessary for the coefficients in homology and cohomology (see the proof of Proposition 3.9).

2.3 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 $F_{\sigma }$ on the Hilbert does not satisfy the Palais–Smale condition, it satisfies a stronger property on a suitable neighbourhood of critical points.

If $(X,\Vert \,\cdot \,\Vert )$ is a Finsler manifold equipped with its Palais distance d and $A\subset X$, we recall the notations

$$\begin{aligned} U_{\delta }(A)=X\cap {\left\{ x: d(x,A)<\delta \right\} },\quad N_{\delta }(A)=X\cap {\left\{ x:d(x,A)\le \delta \right\} }. \end{aligned}$$

Notice in particular that by Corollary 2.4, if A is assumed to be compact, then $N_{\delta }(A)$ is closed and $U_{\delta }(A)$ is its interior. In all constructions, we will assume implicitly whenever necessary that such $\delta >0$ has been chosen such that $N_{\delta }(A)$ is closed.

Proposition 2.16

(Solimini [34]) Let X be a $C^2$ Finsler–Hilbert manifold and $F:X\rightarrow {\mathbb {R}}$ be a $C^2$ function, and assume that $K\subset K(F)$ is a compact subset of set of critical points of F. If the square gradient $\nabla ^2F(x):T_xX\rightarrow T_xX$ is a Fredholm operator for all $x\in X$, for all $\varepsilon >0$, there exists $\delta >0$ such that for all ${\widetilde{F}}:X\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \Vert F-{\widetilde{F}}\Vert _{C^2}\le \varepsilon \end{aligned}$$

the map $\nabla {\widetilde{F}}$ is proper on $N_{\delta }(K)$. In particular, ${\widetilde{F}}$ satisfies the Palais–Smale condition on $N_{\delta }(K)$.

Remark 2.17

That DF is proper near critical points $x\in X$ where $D^2F(x)$ is Fredholm is a well-known property due to Smale [33].

Proof

We first treat the case $K={\left\{ x_0\right\} }$. By a remark which will be repeatedly used, we can assume by Henderson’s theorem [9] that X is an open subset of a Hilbert space H. We fix some $\varepsilon >0$, and we take $\delta >0$ small enough such that $\nabla F-\nabla ^2F(x):X\rightarrow H$ is Lipschitzian on $N_{\delta }(x_0)$ with

$$\begin{aligned} \mathrm {Lip}\left( (\nabla F-\nabla ^2F(x))|N_{\delta }({\left\{ x_0\right\} })\right) \le \varepsilon , \end{aligned}$$

(2.2)

and define $G:X\rightarrow H$ by

$$\begin{aligned} G(x)=\nabla {\widetilde{F}}(x)-\nabla ^2 F(x_0)(x). \end{aligned}$$

Then G is Lipschitzian and satisfies by (2.2)

$$\begin{aligned} \mathrm {Lip}(G|N_{\delta }({\left\{ x_0\right\} }))\le 2\varepsilon . \end{aligned}$$

(2.3)

As $D^2 F(x_0)$ is a Fredholm operator, there exists a finite dimensional vector $H_0=\mathrm {Ker}(\nabla ^2 F(x))\subset H$ such that we have the direct sum decomposition

$$\begin{aligned} H=H_0\oplus H_0^{\perp }. \end{aligned}$$

(2.4)

In particular, as $H_0$ is finite dimensional, $H_0^{\perp }$ is closed and $\mathrm {Im}\,(D^2F(x_0))$ is closed, so there exists a positive constant $0<\alpha <\infty$ such that for all $v\in H_0^{\perp }$, there holds

$$\begin{aligned} \Vert \nabla ^2 F(x_0)(v)\Vert \ge \alpha \Vert v\Vert . \end{aligned}$$

(2.5)

Now, assume that ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset N_{\delta }(x_0)\subset X$ is such that $\{\nabla {\widetilde{F}}(x_k)\}_{k\in {\mathbb {N}}}\subset X$ converges. Writing for all $k\in {\mathbb {N}}$

$$\begin{aligned} x_k=x_k^{0}+x_k^{\perp } \end{aligned}$$

according to the direct sum decomposition (2.4), we can assume that up to subsequence ${\left\{ x_k^{0}\right\} }_{k\in {\mathbb {N}}}$ is convergent in $N_{\delta }(x_0)$ (which is closed by Corollary 2.4). Now, for all $k,l\in {\mathbb {N}}$, we have

$$\begin{aligned} \Vert x_k^{\perp }-x_l^{\perp }\Vert&\le \frac{1}{\alpha }\Vert \nabla ^2F(x_0)(x_k^{\perp }-x_l^{\perp })\Vert =\frac{1}{\alpha }\Vert \nabla ^2F(x_0)(x_k^{\perp }-x_l^{\perp })\Vert \\&\le \frac{1}{\alpha }\Vert G(x_k^{\perp }-x_l^{\perp })\Vert +\frac{1}{\alpha }\Vert \nabla {\widetilde{F}}(x_k^{\perp }-x_l^{\perp })\Vert \\&\le \frac{2\varepsilon }{\alpha }\left( \Vert x_k^{\perp }-x_l^{\perp }\Vert +\Vert x_k^0-x_l^0\Vert \right) +\frac{1}{\alpha }\Vert \nabla {\widetilde{F}}(x_k-x_l)\Vert \end{aligned}$$

where we used (2.3). Therefore, taking $2\varepsilon < \alpha$ yields

$$\begin{aligned} \Vert x_k^{\perp }-x_l^{\perp }\Vert \le \frac{1}{\alpha -2\varepsilon }\left( 2\varepsilon \Vert x_k^{0}-x_l^{0}\Vert +\Vert \nabla {\widetilde{F}}(x_k)-\nabla {\widetilde{F}}(x_l)\Vert \right) {\underset{k,l\rightarrow \infty }{\longrightarrow }}0, \end{aligned}$$

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 $\alpha$ such that (2.5) holds for all $x_0\in K$ and appropriate $H_0=H_0(x_0)$. Taking a finite covering ${\left\{ N_{\delta }(x_i)\right\} }_{1\le i\le N}$ for $\delta >0$ small enough and some elements ${\left\{ x_i\right\} }_{1\le i\le N}\subset K$, the previous proof works identically. This concludes the proof of the general case. $\square$

Corollary 2.18

Let X be a $C^2$ Finsler manifold, $Y\subset X$ be a locally Lipschitz embedded Finsler–Hilbert manifold $F,G\in C^2(X,{\mathbb {R}}_+)$ and for all $0<\sigma <1$, define $F_{\sigma }=F+\sigma ^2G\in C^2(X,{\mathbb {R}}_+)$. Let $\sigma >0$ be a fixed real number and assume that $K\subset K(F_{\sigma })$ is a compact subset such that restriction $\nabla ^2 F_{\sigma }(x):T_xX\rightarrow T_xX$ on X is a Fredholm operator on a compact subset $K\subset K(F_{\sigma })$ (where we recall that $K(F_{\sigma })\subset Y$). Then for all $\varepsilon >0$, there exists $\delta >0$ such that for all ${\widetilde{F}}_{\sigma }\in C^2(X,{\mathbb {R}})$ such that

$$\begin{aligned} \Vert F_{\sigma }-{\widetilde{F}}_{\sigma }\Vert _{C^2} \le \varepsilon \end{aligned}$$

then $\nabla {\widetilde{F}}_{\sigma }:Y\rightarrow Y$ is proper on $N_{\delta }(K)$. In particular, ${\widetilde{F}}$ satisfies the Palais–Smale condition on $N_{\delta }(K)$.

Lemma 2.19

Let X be a Finsler–Hilbert manifold and $K\subset X$ be a compact subset. Then for all small enough $\delta >0$ there exists a smooth function $\varphi :H\rightarrow [0,1]$ whose all derivatives are bounded and such that

$$\begin{aligned} \left\{ \begin{array}{ll} \varphi (x)=1&{}\quad \text {for all}\;\; x\in N_{\delta }(K)\\ \varphi (x)=0&{}\quad \text {for all}\;\; x\in H{\setminus } N_{2\delta }(K). \end{array}\right. \end{aligned}$$

Proof

As K is compact, let $x_1,\ldots ,x_n\in K$ be such that

$$\begin{aligned} K\subset \bigcup _{i=1}^nB\left( x_i,\frac{\delta }{2}\right) . \end{aligned}$$

Taking $\delta$ small enough, we can make sure that each ball $B(x_i,\delta )$ is included in a chart domain into the fixed Hilbert space $(H,|\,\cdot \,|)$ model of X. Let $\eta \in C^{\infty }_c({\mathbb {R}},[0,1])$ be such that

$$\begin{aligned} \left\{ \begin{array}{llll} \eta (t)=1&{}\quad \text {for}\;\, t\le \frac{9}{4}\\ \eta (t)=0&{}\quad \text {for}\;\, t\ge 4 \end{array}\right. \end{aligned}$$

and let $\varphi _i\in C^{\infty }(X,{\mathbb {R}})$ be defined by

$$\begin{aligned} \varphi _i(x)=\eta \left( \frac{|x-x_i|^2}{\delta ^2}\right) . \end{aligned}$$

Then $\varphi _i\in C^{\infty }(H,[0,1])$ verifies

$$\begin{aligned} \left\{ \begin{array}{llll} \varphi _i(x)=1&{}\quad \text {for}\;\, x\in B\left( x_i,\frac{3}{2}\delta \right) \\ \varphi _i(x)=0&{}\quad \text {for}\;\, x\in H{\setminus } B(x_i,2\delta ). \end{array}\right. \end{aligned}$$

Now, letting $\zeta \in C^{\infty }_c({\mathbb {R}},[0,1])$ be such that

$$\begin{aligned} \left\{ \begin{array}{llll} \zeta (t)=0,&{}\quad \text {for}\;\, t\le 0\\ \zeta (t)=1&{}\quad \text {for}\;\, t\ge 1 \end{array}\right. \end{aligned}$$

the function $\varphi \in C^{\infty }(H,[0,1])$ defined by

$$\begin{aligned} \varphi (x)=\zeta \left( \sum _{i=1}^{n}\varphi _i(x)\right) ,\quad x\in H \end{aligned}$$

has all the required properties. $\square$

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 [34]) Let $k\ge 2$ and X be a $C^k$ Finsler–Hilbert manifold and $F:X\rightarrow {\mathbb {R}}$ be a $C^k$ function, and assume that $K_0\subset K=K(F)$ is a compact subset of set of critical points of F. If the square second derivative $\nabla ^2F(x):T_xX\rightarrow T_xX$ is a Fredholm operator for all $x\in K$, then for all $\varepsilon ,\delta >0$ small enough, there exists ${\widetilde{F}}\in C^k(X,{\mathbb {R}})$ such that

$$\begin{aligned}&\Vert F-{\widetilde{F}}\Vert _{C^k(X)}< \varepsilon \nonumber \\&F(x)={\widetilde{F}}(x)\;\, \text {for all}\;\, x\in X{\setminus } N_{2\delta }(K) \end{aligned}$$

(2.6)

and the critical points of ${\widetilde{F}}$ in $N_{\delta }(K)$ are non-degenerate and finite in number. Furthermore, we can impose ${\widetilde{F}}\le F$ or ${\widetilde{F}}\ge F$.

Proof

We can assume by Henderson’s theorem [9] that X is an open subset of a Hilbert space H with scalar product $\langle \,\cdot \,,\,\cdot \,\rangle$. Let $\varphi :X\rightarrow {\mathbb {R}}$ be the cut-off function of Lemma 2.19, and define for $x_0\in N_{2\delta }(K_0)$, $y\in H$ the function ${\widetilde{F}}_{x_0,y}:X\rightarrow {\mathbb {R}}$ such that for all $x\in X$,

$$\begin{aligned} {\widetilde{F}}_{x_0,y}(x)=F(x)+\varphi (x)\langle y,x-x_0\rangle . \end{aligned}$$

As $K_0$ is compact, there exists $C_0=C_0(\delta )$ such that

$$\begin{aligned} \sup _{x\in N_{2\delta }(K_0)}\Vert x\Vert \le C_0(\delta ). \end{aligned}$$

(2.7)

Furthermore, thanks to the construction of Lemma 2.19, we have for some universal constant $C_1$

$$\begin{aligned} \Vert \varphi \Vert _{C^k(X)}=\Vert \varphi \Vert _{C^k(N_{2\delta }(K_0))}\le \frac{C_1}{\delta ^k}. \end{aligned}$$

(2.8)

Then for all $\Vert y\Vert \le \frac{\delta ^k}{C_0(\delta )C_1}\varepsilon$, we get the the first property of (2.6). Furthermore, we have on $N_{\delta }(K_0)$

$$\begin{aligned}&\nabla {\widetilde{F}}_{x_0,y}(x)=\nabla F(x)+y\\&\nabla ^2{\widetilde{F}}_{x_0,y}(x)=\nabla ^2F(x). \end{aligned}$$

In particular, $x\in N_{\delta }(K_0)$ is a non-degenerate critical point of ${\widetilde{F}}_{x,y}$ if $-y$ is a regular value of $\nabla {F}:X\rightarrow H$. Let $\delta >0$ small enough such that each connected component of $N_{\delta }(K_0)$ intersects $K_0$. Let $x\in N_{\delta }(K_0)$. Since X is an open set of the Hilbert space H, notice that $T_xX=H$. By the connectedness assumption, $\nabla ^2F(x):H\rightarrow H$ is Fredholm and as coming from the Hessian of F, $\nabla ^2F(x)!H\rightarrow H$ is a self-adjoint operator. Its Fredholm index is then 0. In particular, we can apply the Sard–Smale theorem on the $C^1$ map $\nabla F:N_{\delta }(K_0)\rightarrow H$ (Theorem 2.7).

We obtain an element $-y\in X$ such that

$$\begin{aligned} \Vert y\Vert < \frac{\delta ^k}{C_0(\delta )C_1}\varepsilon \end{aligned}$$

(2.9)

and such that $-y$ is a regular value of $\nabla F_{x_0,y}$ (for all $x_0\in X$). Writing ${\widetilde{F}}_{x_0}={\widetilde{F}}_{x_0,y}$, we see that for all $x_0\in N_{2\delta }(K_0)$, by (2.7), (2.8) and (2.9)

$$\begin{aligned} \Vert F-{\widetilde{F}}_{x_0,y}\Vert _{C^k(X)}< \varepsilon \end{aligned}$$

Now, once $y\in X$ is chosen, as $K_0$ is compact,

$$\begin{aligned} \sup _{x\in N_{2\delta }(K_0)}\langle y,x\rangle <\infty , \end{aligned}$$

and there exists $x_0\in N_{2\delta }(K_0)$ such that

$$\begin{aligned} \langle y,x\rangle \le \langle y,x_0\rangle \quad \text {for all}\;\, x\in N_{2\delta }(K_0). \end{aligned}$$

Taking ${\widetilde{F}}={\widetilde{F}}_{x_0,y}$, we obtain ${\widetilde{F}}\le {F}$ and the conclusions of the Proposition (the other inequality ${\widetilde{F}}\ge F$ is similar), using Proposition 2.16: the Palais–Smale condition and non-degenerateness implies that the number of critical points is finite. $\square$

3 Lazer–Solimini deformation theorem

3.1 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 [34]).

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 space and a $C^2$ function $F:X\rightarrow {\mathbb {R}}$, for which we assume that F satisfies the Palais–Smale condition at all level $c\in {\mathbb {R}}$, and to fix ideas, let ${\mathscr {A}}$ be a d-dimensional admissible family. We assume that F is non-degenerate on the critical set $K(F,\beta _0)$ at level $\beta _0=\beta (F,{\mathscr {A}})$. In particular, as F satisfies the Palais–Smale condition, $K(F,\beta _0)$ is compact and as F is non-degenerate on $K(F,\beta _0)$, we deduce that $K(F,\beta _0)$ is composed of finitely many points, so that for some $x_1,\ldots ,x_m\in X$, we have

$$\begin{aligned} K(F,\beta _0)={\left\{ x_1,\ldots ,x_m\right\} }. \end{aligned}$$

Let $i\in {\left\{ 1,\ldots ,m\right\} }$ be a fixed integer. Then there exists closed subspaces $H_-,H_+\subset H$ such that up to the identification $T_{x_i}X\simeq H$, the square gradient $\nabla ^2F(x)\in {\mathscr {L}}(T_{x_i}X)$ is negative definite on $H_-$ and positive definite on $H_+$. Furthermore, H is the direct sum of $H_-$ and $H_+$, and for all $y\in H=H_-\oplus H_+$, we write $y=y_-+y_+$, where $y_-\in H_-$ and $y_+\in H_+$.

Furthermore, by the Morse lemma for $C^2$ functions [4], for all $1\le i\le m$, there exists $\varepsilon _i>0$ and a Lipschitzian homeomorphism $\varphi _i:U_{\varepsilon _i}(x_i)\rightarrow \varphi (U_{\varepsilon _i}(x_i))\subset H$ such that $\varphi _i(x_i)=0\in H$ and for all $x\in \varphi (U_{\varepsilon }(x_i))$, there holds

$$\begin{aligned} F(\varphi _i^{-1}(x))=F(x_i)+\Vert x_+\Vert ^2-\Vert x_-\Vert ^2, \end{aligned}$$

(3.1)

where $\Vert \,\cdot \,\Vert$ 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 $B_-(0,r_1)\subset H_-$ and $B_+(0,r_2)\subset H_+$ such that

$$\begin{aligned} B_-(0,2r_1)+B_+(0,r_2)\subset \varphi (U_{\varepsilon _i}(x_i)). \end{aligned}$$

Now, we define for all $0<s\le 2r_1$ and $0<t\le r_2$

$$\begin{aligned} C(s,t)=\varphi ^{-1}(B_-(0,s)+B_+(0,t))\subset U_{\varepsilon _i}(x_i)\subset X. \end{aligned}$$

Now, fix $0<\delta <r_2^2-4r_1^2$, and let $\zeta :{\mathbb {R}}\rightarrow [0,1]$ be a smooth cut-off function such that $\mathrm {supp}(\zeta )\subset {\mathbb {R}}_+$ and $\zeta (t)=1$ for all $t\ge 1$. Now, we define a map $\Phi :X\rightarrow X$ such that

$$\begin{aligned} \begin{aligned}&\Phi (x)=x\quad&\text {for all}\;\, x\in X{\setminus } C(2r_1,r_2)\\&\Phi (x)=\varphi ^{-1}\left( \zeta \left( \frac{\Vert \varphi (x)_-\Vert }{r_1}-1\right) \varphi (x)_++\varphi (x)_-\right) \quad&\text {for all}\;\, x\in C(2r_1,r_2). \end{aligned} \end{aligned}$$

Lemma 3.1

The map $\Phi :X\rightarrow X$ is continuous on $X{\setminus } \varphi ^{-1}(B_-(0,2r_1)+ \partial B_+(0,r_2))$,

$$\begin{aligned} X\cap {\left\{ x:F(x)\le \beta _0+\delta \right\} }\subset X{\setminus } \varphi ^{-1}(B_-(0,2r_1)+\partial B_+(0,r_2)), \end{aligned}$$

(3.2)

and the function $\Phi$ is Lipschitzian on $X\cap {\left\{ x: F(x)\le \beta _0+\delta \right\} }$. Furthermore, it satisfies to the following properties:

• (1) For all $x\in X$, then $F(\Phi (x))\le F(x)$.

• (2) If $x\in \partial C(r_1,r_2)$ and $F(x)\le F(x_i)+\delta$, then $\Phi (x)\in \varphi ^{-1}(\partial B_-(0,r_1))$.

Proof

To check (3.2), it suffices by taking complements to show that for all $x\in \varphi ^{-1}(B_-(0,2r_1)+\partial B_+(0,r_2))$, we have

$$\begin{aligned} F(x)>\beta _0+\delta . \end{aligned}$$

For all $x\in \varphi ^{-1}(B_-(0,2r_1)+\partial B_+(0,r_2))$, we have $\Vert \varphi (x)_+\Vert =r_2$, and $\Vert \varphi (x)_-\Vert \le 2r_1$, so that by (3.1)

$$\begin{aligned} F(x)=\beta _0+\Vert \varphi (x)_+\Vert ^2-\Vert \varphi (x)_-\Vert ^2=\beta _0+r_2^2-\Vert \varphi (x)_-\Vert ^2\ge \beta _0+r_2^2-4r_1^2>\beta _0+\delta \end{aligned}$$

by definition of $0<\delta <r_2^2-4r_1^2$, which shows the claim.

• (1) As $\Phi =\mathrm {Id}$ on $X{\setminus } C(2r_1,r_2)$, it suffices to check the property on $C(2r_1,r_2)$. If $x\in C(2r_1,r_2)$, then by (3.1) and as $\zeta \le 1$

  $$\begin{aligned} F(\Phi (x))&=F\left( \varphi ^{-1}\left( \zeta \left( \frac{\Vert \varphi (x)_-\Vert }{r_1}-1\right) \varphi (x)_++\varphi (x)_-\right) \right) \\&=F(x_i)+\zeta \left( \frac{\Vert \varphi (x)_-\Vert }{r_1}-1\right) ^2\Vert \varphi (x)_+\Vert ^2-\Vert \varphi (x)_-\Vert ^2\\&\le F(x_i)+\Vert \varphi (x)_+\Vert ^2-\Vert \varphi (x)_-\Vert ^2=F(x). \end{aligned}$$

• (2) If $x\in \partial C(r_1,r_2)$ and $F(x)\le F(x_i)+\delta$, recalling that $0<\delta <r_2^2-4r_1^2$, we see that

  $$\begin{aligned} F(x)=F(x_i)+\Vert \varphi (x)_+\Vert ^2-\Vert \varphi (x)_-\Vert ^2<F(x_i)+r_2^2-4r_1^2. \end{aligned}$$

  Therefore, as $x\in \partial C(r_1,r_2)$ we must have $\Vert \varphi (x)_-\Vert =r_1$, so that

  $$\begin{aligned} \zeta \left( \frac{\Vert \varphi (x)_-\Vert }{r_1}-1\right) =0 \end{aligned}$$

  and $\Phi (x)=\varphi ^{-1}\left( \varphi (x)_-\right)$, and as $\Vert \varphi (x)_-\Vert =r_1$, this exactly means that $\Phi (x)\in \varphi ^{-1}(\partial B_-(0,r_1))$. $\square$

Remark 3.2

It is also claimed (without proof, which is left to the reader) in [8, 12] that we have the additional property:

• (3) We have $\Phi (X{\setminus } C(r_1,r_2))\subset X{\setminus } C(r_1,r_2)$.

As $\Phi =\mathrm {Id}$ on $X{\setminus } C(2r_1,r_2)$, we indeed have trivially

$$\begin{aligned} \Phi (X{\setminus } C(2r_1,r_2))=X{\setminus } C(2r_1,r_2)\subset X{\setminus } C(2r_1,r_2). \end{aligned}$$

Therefore, the property is equivalent to

$$\begin{aligned} \Phi \left( C(2r_1,r_2){\setminus } \mathrm {int}(C(r_1,r_2))\right) \subset X{\setminus } \mathrm {int}(C(r_1,r_2)). \end{aligned}$$

Let $x\in C(2r_1,r_2){\setminus } \mathrm {int}(C(r_1,r_2))$ be a fixed element. Then at least one of the properties $r_1\le \Vert \varphi (x)_+\Vert \le 2r_1$ or $\Vert \varphi (x)_+\Vert =r_2$ holds. Furthermore, as

$$\begin{aligned} \varphi (\Phi (x))=\zeta \left( \frac{\Vert \varphi (x)_+\Vert }{r_1}-1\right) \varphi (x)_++\varphi (x)_-, \end{aligned}$$

we trivially obtain

$$\begin{aligned} \varphi (\Phi (x))_+=\zeta \left( \frac{\Vert \varphi (x)_+\Vert }{r_1}-1\right) \varphi (x)_+,\quad \varphi (\Phi (x))_-=\varphi (x)_-. \end{aligned}$$

Therefore, $\Phi (x)\in \mathrm {int}(C(r_1,r_2))=U_{-}(0,r_1)+U_+(0,r_2)$ if and only if

$$\begin{aligned} \Vert \varphi (\Phi (x))_+\Vert =\zeta \left( \frac{\Vert \varphi (x)_-\Vert }{r_1}-1\right) \Vert \varphi (x)_+\Vert<r_2\quad \text {and}\quad \Vert \varphi (\Phi (x))_-\Vert =\Vert \varphi (x)_-\Vert <r_1. \end{aligned}$$

(3.3)

The second inequality in (3.3) implies that $\Vert \varphi (x)_-\Vert <r_1$, so $\Vert \varphi (x)_+\Vert = r_2$ (as $x\in C(2r_1,r_2){\setminus } \mathrm {int}(C(r_1,r_2))$, and (3.3) is equivalent to

$$\begin{aligned} \zeta \left( \frac{\Vert \varphi (x)_-\Vert }{r_1}-1\right) <1, \end{aligned}$$

and by construction of $\zeta$, we see that there exists $0<\delta <1$ such that

$$\begin{aligned} \zeta \left( \frac{t}{r_1}-1\right)<1\;\, \text {for all}\;\, t<\delta (2r_1). \end{aligned}$$

This implies that

$$\begin{aligned} \Phi (\varphi ^{-1}(B_-(0,\delta (2r_1))+\partial B_+(0,r_2)))\subset \mathrm {int}(C(r_1,r_2)) \end{aligned}$$

and as trivially

$$\begin{aligned} \varphi ^{-1}(B_-(0,\delta ( 2r_1))+\partial B_+(0,r_2))\not \subset \mathrm {int}(C(r_1,r_2))=\varphi ^{-1}\left( U_-(0,r_1)+U_+(0,r_2)\right) , \end{aligned}$$

(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.

Lemma 3.3

Let K be a closed set in a d-dimensional $C^1$ manifold $M^d$ and H be a Hilbert space and let $f:K\rightarrow H$ be a continuous function such that $0\notin f(K)$. If $d<\dim H$, there exists a continuous extension ${\overline{f}}:M^d\rightarrow H{\setminus }{\left\{ 0\right\} }$.

Proof

First assume that K is compact, and let $r>0$ such that $K\subset B(0,r)$. Then we obtain an extension ${\overline{f}}:M^d\rightarrow H$ 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 ${\overline{f}}|_{M^d{\setminus } K}:M^d{\setminus } K\rightarrow H$ is $C^1$. In particular, as ${\overline{f}}|_{M^d{\setminus } K}:M^d{\setminus } K\rightarrow H$ is locally Lipschitzian,

$$\begin{aligned} \mathrm {dim}_{{\mathscr {H}}}({\overline{f}}(M^d{\setminus } K))\le d<\mathrm {dim}\, H, \end{aligned}$$

(3.5)

where $\mathrm {dim}_{{\mathscr {H}}}$ designs the Hausdorff measure of the metric space H induced with its natural distance. In particular, as $0\notin {\overline{f}}(K)=f(K)$ by assumption, and as ${\overline{f}}(M^d{\setminus } K)$ cannot contain an open ball by (3.5) (otherwise it would be of Hausdorff dimension $\dim H\ge d+1$), we deduce that $B(0,r)\not \subset {\overline{f}}(M^d)$. In particular, if $x_0\in B(0,r){\setminus } {\overline{f}}(M^d)$ is a fixed point, we can set $p:{\overline{f}}(M^d)\cap B(0,r)\rightarrow \partial B(0,r)$ defined by $p(x)=x_0+\alpha (x-x_0)$, where $\alpha$ is the unique positive number such that $\Vert p(x)\Vert =r$.

If K is not compact, we fix some arbitrary point $p\in M^d$ and for all $n\in {\mathbb {N}}$, we let $K_n=K\cap {\overline{B}}(p,n)$. We apply the previous construction to the restriction $f_{K_1}:K_1\rightarrow H{\setminus }{\left\{ 0\right\} }$ to obtain an extension ${\overline{f}}_{K_1}:M^d\rightarrow H{\setminus }{\left\{ 0\right\} }$ . Now, let ${\overline{f}}_1:{\overline{B}}(p,1)\cup K\rightarrow H{\setminus }{\left\{ 0\right\} }$ be the extension by f on $K{\setminus } {\overline{B}}(p,1)$ of the restriction ${{\overline{f}}_{K_1}}_{|{\overline{B}}(p,1)}:{\overline{B}}(p,1)\rightarrow H{\setminus }{\left\{ 0\right\} }$. This gives a family of functions $f_n:{\overline{B}}(p,n)\cup K\rightarrow H{\setminus }{\left\{ 0\right\} }$ such that for all $m\ge n$, $f_n=f_m$ on ${\overline{B}}(p,n)\cup K$, so all these functions have a common extension ${\overline{f}}:M^d\rightarrow H{\setminus }{\left\{ 0\right\} }$, and this concludes the proof of the Lemma. $\square$

Remark 3.4

As, we only use the Lipschitz property, the proof would carry one to metric spaces of Hausdorff dimension at most d admitting Lipschitzian partitions of unity—in particular, this would work for Lipschitz manifolds (notice that the part using Dugundji extension theorem works for any metric space). More generally, we observe that the Lemma would still hold if the function ${\overline{f}}|_{M^d{\setminus } K}:M^d$ was only $\alpha$-Hölder with $\alpha >\dfrac{d}{d+1}$, so we could relax the hypotheses to metric spaces admitting $\alpha$-Hölder partitions of unity.

3.2 The index bounds for non-degenerate functions on Finsler–Hilbert manifolds

Definition 3.5

If ${\mathscr {A}}$ is a min–max family and $F\in C^1(X,{\mathbb {R}})$, and ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset {\mathscr {A}}$ is such that

$$\begin{aligned} \sup F(A_k){\underset{k\rightarrow \infty }{\longrightarrow }}\beta (F,{\mathscr {A}})=\inf _{A\in {\mathscr {A}}}\sup F(A), \end{aligned}$$

we define

$$\begin{aligned} A_{\infty }=X\cap {\left\{ x: x=\lim \limits _{k\rightarrow \infty }x_k,\;\, \mathrm {dist}(x_k,A_k){\underset{k\rightarrow \infty }{\longrightarrow }}0\right\} }. \end{aligned}$$

Theorem 3.6

(Lazer–Solimini [12]) Let X be a $C^2$ Finsler–Hilbert manifold, ${\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, resp. $\widetilde{{\mathscr {A}}}$) be a d-dimensional admissible family (resp. dual family, resp. co-dual family) with boundary ${\left\{ C_i\right\} }_{i\in I}\subset X$ and let $F\in C^2(X,{\mathbb {R}})$ be such that F satisfies the Palais–Smale at level $\beta _0=\beta (F,{\mathscr {A}})$. Assume furthermore that all critical points of F are non-degenerate at level $\beta _0$, and that the min–max is non-trivial, i.e.

$$\begin{aligned} \beta _0&=\inf _{A\in {\mathscr {A}}}\sup F(A)>\sup _{i\in I}\,\sup F(C_i)={\widehat{\beta }}_0\\ \Big (\text {resp. } \beta _0^{*}&=\inf _{A\in {\mathscr {A}}^{*}}\sup F(A)>\sup _{i\in I}\,\sup F(C_i)={\widehat{\beta }}_0\Big )\\ \Big (\text {resp. } {\widetilde{\beta }}_0&=\inf _{A\in \widetilde{{\mathscr {A}}}}\sup F(A)>\sup _{i\in I}\,\sup F(C_i)={\widehat{\beta }}_0\Big ). \end{aligned}$$

Then for all ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset {\mathscr {A}}$ such that

$$\begin{aligned}&\sup F(A_k){\underset{k\rightarrow \infty }{\longrightarrow }}\beta _0,\\ \Big (\text {resp.}\;\,&\sup F(A_k){\underset{k\rightarrow \infty }{\longrightarrow }}\beta _0^{*} \Big )\\ \Big (\text {resp.}\;\,&\sup F(A_k){\underset{k\rightarrow \infty }{\longrightarrow }}{\widetilde{\beta }}_0\Big ) \end{aligned}$$

the exists $x\in K(F,\beta _0)$ (resp. $x^{*}\in K(F,\beta _0^{*})$, resp. ${\widetilde{x}}\in K(F,{\widetilde{\beta }}_0)$) and a sequence ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that $x_k\in A_k$ for all $k\in {\mathbb {N}}$, $x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x$ (resp. $x^{*}$, resp. ${\widetilde{x}}$) and

$$\begin{aligned} \mathrm {Ind}_{F}(x)\le d,\quad (\text {resp.}\;\, \mathrm {Ind}_{F}(x^{*})\ge d,\;\, \text {resp.}\;\, \mathrm {Ind}_{F}({\widetilde{x}})=d). \end{aligned}$$

Remark 3.7

The proof shows that it suffices to assume that F is non-degenerate on $K(F,\beta _0)\cap A_{\infty }$.

This is easy to see that the proof is reduced to the following Theorem (from it one obtains immediately Theorem (3.6), as we shall see shortly).

Proposition 3.8

(Lazer–Solimini [12], Solimini, Lemma 2.19 [34]) Let $F\in C^2(X,{\mathbb {R}}_+)$ as in Theorem 3.6, let ${\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, or $\overline{{\mathscr {A}}}$) be a d-dimensional admissible min–max family and assume that all critical points at level $\beta _0$ (resp. at level $\beta ^{*}_0$, or ${\widetilde{\beta }}_0$) are non-degenerate, and assume that $x_0\in K(F,\beta _0)$ (resp. $x_0\in K(F,\beta ^{*}_0)$, resp. ${x}_0\in K(F,{\widetilde{\beta }}_0)$) satisfies the estimate

$$\begin{aligned} \mathrm {Ind}_{F}(x_0)>d \end{aligned}$$

(3.6)

respectively for ${\mathscr {A}}^{*}$

$$\begin{aligned} \mathrm {Ind}_{F}(x_0) <d \end{aligned}$$

(3.7)

and for $\widetilde{{\mathscr {A}}}$

$$\begin{aligned} \mathrm {Ind}_{F}(x_0) \ne d. \end{aligned}$$

(3.8)

Then for all small enough $\varepsilon >0$, there exists $\delta >0$ such that for all $A\in {\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, or $\overline{{\mathscr {A}}}$), $\sup F(A)\le \beta _0+\delta$ (resp. $\sup F(A)\le \beta _0^{*}+\delta$, resp. $\sup F(A)\le {\widetilde{\beta }}_0+\delta$ ) implies that there exists $A'\in {\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, or $\overline{{\mathscr {A}}}$) such that

$$\begin{aligned} \left\{ \begin{array}{ll} A{\setminus } U_{2\varepsilon }(x_0)=A'{\setminus } U_{2\varepsilon }(x_0)\\ A'\cap U_{\varepsilon }(x_0)=\varnothing \\ \sup F(A')\le \sup F(A). \end{array}\right. \end{aligned}$$

(3.9)

Proof

Case 1: admissible families.

Taking the previous notations of Lemma 3.1, we will show that for $0<\delta <r_2^2-4r_1^2$, there exists $A'\in {\mathscr {A}}$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} A'{\setminus } C(2r_1,r_2)=A{\setminus } C(2r_1,r_2)\\ A'\cap \mathrm {int}(C(r_1,r_2))=\varnothing \\ \sup F(A')\le \sup F(A)\le \beta _0+\delta . \end{array}\right. \end{aligned}$$

(3.10)

First, let $f\in C^0(M^d,X)$ such that $A=f(M^d)$, and consider the open subset $U=f^{-1}(\mathrm {int}(C(r_1,r_2)))\subset M^d$. For all $p\in \partial U=f^{-1}(\partial C(r_1,r_2))\subset M^d$, we have by definition $f(p)\in \partial C(r_1,r_2)$, and

$$\begin{aligned} F(f(p))\le \sup F(A)\le \beta _0+\delta , \end{aligned}$$

so by Lemma 3.1 (2), we have $\Phi (f(p))\in \varphi ^{-1}(\partial B_-(0,r_1))$. As $p\in \partial U$ was arbitrary we obtain

$$\begin{aligned} \Phi (f(\partial U))\subset \varphi ^{-1}(\partial B_-(0,r_1)). \end{aligned}$$

(3.11)

Now, $\varphi :\varphi ^{-1}(B_-(0,r_1))\rightarrow B_-(0,r_1)$ is a (Lipschitzian) homeomorphism, so it induces a homeomorphism on the boundary

$$\begin{aligned} \varphi :\varphi ^{-1}(\partial B_-(0,r_1))\rightarrow \partial B_-(0,r_1). \end{aligned}$$

Furthermore, as $\partial B_-(0,r_1)\subset H_-$ is a retract by deformation of $H_-{\setminus }{\left\{ 0\right\} }$, we see by Lemma 3.3 that $\varphi \circ \Phi \circ f:\partial U\rightarrow \partial B_-(0,r_1)\subset H_-{\setminus }{\left\{ 0\right\} }$ can be extended as a map $\Psi :{\overline{U}}\rightarrow \partial B_-(0,r_1)$ (by using the projection $H_-{\setminus }{\left\{ 0\right\} }\rightarrow \partial B_-(0,r_1)$), and the map $\overline{\Phi \circ f}=\varphi ^{-1}\circ \Psi :{\overline{U}}\rightarrow \varphi ^{-1}(\partial B_-(0,r_1))$ furnishes a continuous extension of $\Phi \circ f:\partial U\rightarrow \varphi ^{-1}(\partial B_-(0,r_1))$. Now, define the continuous map ${\widetilde{f}}:M^d\rightarrow X$ by

$$\begin{aligned} {\widetilde{f}}(p)= \left\{ \begin{array}{llll} f(p)\quad &{}\quad \text {for all}\;\, p\in M^d{\setminus } {\overline{U}},\\ \overline{\Phi \circ f}(p)&{}\quad \text {for all}\;\, p\in {\overline{U}}. \end{array}\right. \end{aligned}$$

We first need to check that $A'={\widetilde{f}}(M^d)$ satisfies the non-triviality of the boundary condition. First, up to taking $r_1,r_2>0$ smaller, as

$$\begin{aligned} F(x_0)=\beta _0>\sup _{i\in I}\sup F(h_i(B_i^{d-1})) \end{aligned}$$

we can assume that $C(2r_1,r_2)\cap h_i(B_i^{d-1})=\varnothing$ as F is continuous. In particular, as $\Phi =\mathrm {Id}$ on $X{\setminus } C(2r_1,r_2)$, we have ${\widetilde{f}}|_{B_i^{d-1}}=f_{|B^{d-1}}$ on $B_i^{d-1}$ for all $i\in I$, so $A'\in {\mathscr {A}}$. Furthermore, for all $p\in M^d{\setminus } f^{-1}(C(2r_1,r_2))$, ${\widetilde{f}}(p)=f(p)$, so

$$\begin{aligned} F({\widetilde{f}}(p))=F(f(p))\le \sup F(A)\le \beta _0+\delta . \end{aligned}$$

Then, for all $p\in f^{-1}(C(2r_1,r_2){\setminus } \mathrm {int}(C(2r_1,r_2)))$, we have by Lemma 3.1(1)

$$\begin{aligned} F({\widetilde{f}}(p))=F(\Phi (f(p)))\le F(f(p))\le \sup F(A)\le \beta _0+\delta , \end{aligned}$$

and finally, for all $p\in f^{-1}(\mathrm {int}(C(r_1,r_2)))=U$, we have by construction ${\widetilde{f}}(p)\in \varphi ^{-1}(\partial B_-(0,r_1))$, but this implies by (3.1) that

$$\begin{aligned} F({\widetilde{f}}(p))=\beta _0-\Vert \varphi ({\widetilde{f}}(p))_-\Vert ^2<\beta _0\le \sup F(A). \end{aligned}$$

Finally, as $A'\cap \mathrm {int}(C(r_1,r_2))=\varnothing$ and $A'{\setminus } C(2r_1,r_2)=A{\setminus } C(2r_1,r_2)$, this proves (3.10).

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 $0<\eta <\delta$ and for all $A\in {\mathscr {A}}^{*}$ such that

$$\begin{aligned} \sup F(A)\le \beta ^{*}_0+\eta , \end{aligned}$$

there holds (notice that $\Phi (A)\in {\mathscr {A}}^{*}$ by construction of $\Phi$)

$$\begin{aligned} A'=\Phi (A){\setminus } \mathrm {int}(C(r_1,r_2))\in {\mathscr {A}}^{*}, \end{aligned}$$

(3.12)

which will immediately imply the claim, as $F(\Phi (x))\le F(x)$ for all $x\in X$, so that

$$\begin{aligned} \sup F(A')=\sup F(\Phi (A))\le \sup F(A)\le \beta ^{*}_0+\eta . \end{aligned}$$

Now assume by contradiction that (3.12) does not hold. This means by Definition 2.9 that there exists a continuous map $h:\Phi (A){\setminus } \mathrm {int}(C(r_1,r_2))\rightarrow {\mathbb {R}}^d{\setminus } {\left\{ 0\right\} }$ such that for some $i\in I$ and $j\in J_i$, we have

$$\begin{aligned} h(x)=h_i^j(x)\quad \text {for all}\;\, x\in C_i\cap \Phi (A). \end{aligned}$$

Now, consider the restriction $h\circ \varphi ^{-1}:\varphi (\Phi (A)\cap \partial C(r_1,r_2))\rightarrow {\mathbb {R}}^d{\setminus }{\left\{ 0\right\} }$. As $\varphi (\Phi (A)\cap \partial C(r_1,r_2))\subset H_-$ and $\mathrm {dim}(H_-)=\mathrm {Ind}_{F}(x_0)<d$, we deduce by Lemma 3.1 that there exists an extension

$$\begin{aligned} \overline{h\circ \varphi ^{-1}}:\varphi (\Phi (A)\cap C(r_1,r_2)\rightarrow {\mathbb {R}}^d{\setminus }{\left\{ 0\right\} }, \end{aligned}$$

and

$$\begin{aligned} {\overline{h}}=\overline{h\circ \varphi ^{-1}}\circ \varphi :\Phi (A)\cap C(r_1,r_2)\rightarrow {\mathbb {R}}^d{\setminus }{\left\{ 0\right\} } \end{aligned}$$

is a continuous extension of $h|_{\Phi (A)\cap \partial C(r_1,r_2)}:\Phi (A)\cap \partial C(r_1,r_2)\rightarrow {\mathbb {R}}^d{\setminus }{\left\{ 0\right\} }$. Finally, if ${\widetilde{h}}:\Phi (A)\rightarrow {\mathbb {R}}^d{\setminus }{\left\{ 0\right\} }$ is the continuous map given by

$$\begin{aligned} {\widetilde{h}}(x)= \left\{ \begin{array}{llll} h(x),&{}\quad \text {for all}\;\, x\in \Phi (A){\setminus } \mathrm {int}(C(r_1,r_2)),\\ {\overline{h}}(x),&{}\quad \text {for all}\;\, x\in \Phi (A)\cap C(r_1,r_2) \end{array}\right. \end{aligned}$$

this implies by definition of ${\mathscr {A}}^{*}$ that $\Phi (A)\notin {\mathscr {A}}^{*}$, a contradiction (as $0\notin \mathrm {Im}({\widetilde{h}})$).

Case 3: co-dual admissible families.

First, the argument of Case 2 shows that we only need to treat the case $\mathrm {Ind}_{F}(x_0)<d$, as the map $\varphi :U_{\varepsilon }(x_0)\rightarrow \varphi (U_{\varepsilon }(x_0))\subset H_-$ is a locally bi-Lipschitzian homeomorphism, so the map $\Phi :X\rightarrow X$ is locally Lipschitzian on A, so that

$$\begin{aligned} \mathrm {dim}_{{\mathscr {H}}}(\Phi (A))\le \mathrm {dim}_{{\mathscr {H}}}(A)<d+1 \end{aligned}$$

(3.13)

and as $\Phi (A)\in {\mathscr {A}}^{*}$, we obtain by (3.13) that $\Phi (A)\in \widetilde{{\mathscr {A}}}$.

Therefore, we see that we can assume that $\mathrm {Ind}_{F}(x_0)=\mathrm {dim}(H_-)\ge d+1$. Once again, as the map $\varphi :U_{\varepsilon }(x_0)\rightarrow \varphi (U_{\varepsilon }(x_0))\subset H_-$ is a locally bi-Lipschitzian homeomorphism, and $\Phi : X\rightarrow X$ is locally Lipschitzian on A, we have

$$\begin{aligned} \mathrm {dim}_{{\mathscr {H}}}\left( \varphi (\Phi (A)\cap C(r_1,r_2))\right) \le \mathrm {dim}_{{\mathscr {H}}}(A)<d+1. \end{aligned}$$

(3.14)

Now, we trivially have by (3.14)

$$\begin{aligned} \mathrm {dim}(B_-(0,r_1))=\mathrm {dim}(U_-(0,r_1))=\mathrm {dim}_{{\mathscr {H}}}(H_-)\ge d+1>\mathrm {dim}_{{\mathscr {H}}}\left( \varphi (\Phi (A)\cap C(r_1,r_2))\right) \end{aligned}$$

(3.15)

In particular, we deduce from (3.15) that

$$\begin{aligned} B_-(0,r_1)\not \subset \varphi (\Phi (A)\cap C(r_1,r_2)). \end{aligned}$$

(3.16)

Now, as $\varphi (\Phi (A)\cap C(r_1,r_2))$ is closed, there exists $\eta >0$ and $x_0\in U_-(0,r_1)$ such that

$$\begin{aligned} B(x_0,\eta )\cap H_-\subset B(0,r_1){\setminus } \varphi (\Phi (A)\cap C(r_1,r_2)). \end{aligned}$$

Furthermore, as the projection $\pi : B_-(0,r_1){\setminus } {\left\{ x_0\right\} }\rightarrow \partial B_-(0,r_1)$ is Lipschitzian outside of $B(x_0,\eta )$, we see that

$$\begin{aligned} A'=(\varphi ^{-1}\circ \pi ) (\varphi (\Phi (A)\cap C(r_1,r_2))\in \widetilde{{\mathscr {A}}} \end{aligned}$$

thanks to (3.14) and as $\varphi ^{-1}\circ \pi$ is locally Lipschitzian on $\varphi (\Phi (A)\cap C(r_1,r_2))$. By definition, we have

$$\begin{aligned} A'\cap \mathrm {int}(C(r_1,r_2))=\varnothing . \end{aligned}$$

We finally check that

$$\begin{aligned} \sup F(A')\le \sup F(A)\le {\widetilde{\beta }}_0+\delta . \end{aligned}$$

By Lemma 3.1, we have

$$\begin{aligned} \sup F(\Phi (A))\le \sup F(A) \end{aligned}$$

(3.17)

and as $A'{\setminus } \Phi (A)\subset \varphi ^{-1}(\partial B_-(0,r_1))$, we obtain by (3.1)

$$\begin{aligned} F(x)={\widetilde{\beta }}_0-\Vert \varphi (x)_-\Vert ^2<{\widetilde{\beta }}_0\le \sup F(A),\quad \text {for all}\;\, x\in A'{\setminus } \Phi (A), \end{aligned}$$

(3.18)

so that by (3.17) and (3.18)

$$\begin{aligned} \sup F(A')\le \sup F(A)\le {\widetilde{\beta }}_0+\delta , \end{aligned}$$

which concludes the proof of the theorem. $\square$

Proof

(of Theorem 3.6) The proof is a reductio ad absurdum. Assume that there is no such critical point. As the conclusions of Proposition 3.8 are independent of the admissible family, we can assume that ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset X$ is such that $A_k\in {\mathscr {A}}$ for all $k\in {\mathbb {N}}$ and

$$\begin{aligned} \sup F(A_k){\underset{k\rightarrow \infty }{\longrightarrow }}\beta _0. \end{aligned}$$

Now, let

$$\begin{aligned} A_{\infty }=X\cap {\left\{ x=\lim \limits _{k\rightarrow \infty }x_k: \;\, \mathrm {dist}(x_k,A_k){\underset{k\rightarrow \infty }{\longrightarrow }}0\right\} }. \end{aligned}$$

(3.19)

Then by assumption, F is non-degenerate on $K(F,\beta _0)\cap A_{\infty }$, and as $K(F,\beta _0)\cap A_{\infty }$ is compact by the Palais–Smale condition, we deduce by the Morse lemma that $K(F,\beta _0)\cap A_{\infty }$ is finite, so we have for some $x_1,\ldots ,x_m\in X$

$$\begin{aligned} K(F,\beta _0)\cap A_{\infty }={\left\{ x_1,\ldots ,x_m\right\} }. \end{aligned}$$

Now, thanks to Proposition 3.8, as $K(F,\beta _0)\cap A_{\infty }$ is finite, there exists $\delta ,\varepsilon >0$ such that for all $A\in {\mathscr {A}}$, $\sup F(A)\le \beta _0+\delta$, there exists for all $1\le i\le m$ an element $A_i'\in {\mathscr {A}}$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} A{\setminus } U_{2\varepsilon }(x_i)=A_i'{\setminus } U_{2\varepsilon }(x_i)\\ A_i'\cap U_{\varepsilon }(x_i)=\varnothing \\ \sup F(A_i')\le \sup F(A). \end{array}\right. \end{aligned}$$

(3.20)

Now, we taking $\varepsilon >0$ sufficiently small, we can assume that

$$\begin{aligned} {\overline{B}}_{2\varepsilon }(x_i)\cap {\overline{B}}_{2\varepsilon }(x_j)=\varnothing ,\quad \text {for all}\;\, i\ne j\;\,\text {with}\;\, i,j\in {\left\{ 1,\ldots ,m\right\} } \end{aligned}$$

(3.21)

Thanks to (3.20) and (3.21), we see that $A_k$ satisfies the hypotheses to obtain (3.20) for k large enough define by a finite induction $A_k^1,\ldots A_k^m\in {\mathscr {A}}$ by

$$\begin{aligned} A_k^1=(A_k)_1',\quad A_k^{i}=(A_k^{i-1})_i,\quad \text {for all}\;\, 2\le i\le m. \end{aligned}$$

Then $A_k^m\in {\mathscr {A}}$ and

$$\begin{aligned} \sup F(A_k^m)\le \sup F(A_{k}){\underset{k\rightarrow \infty }{\longrightarrow }}\beta _0, \end{aligned}$$

so by any deformation lemma (see e.g. [34]), there exists ${\left\{ x_k^m\right\} }_{k\in {\mathbb {N}}}$ such that $x_k^m\in A_k^m$ for all $k\in {\mathbb {N}}$, and

$$\begin{aligned} \mathrm {dist}(x_k^m,{\left\{ x_1,\ldots ,x_m\right\} })\ge \varepsilon ,\quad \text {for all}\;\, k\in {\mathbb {N}}. \end{aligned}$$

and $x_k{\underset{k\rightarrow \infty }{\longrightarrow }} x_{\infty }\in K(F,\beta _0)$. Furthermore, assuming that $\varepsilon >0$ is small enough, and as ${\left\{ x_1,\ldots ,x_m\right\} }=K(F,\beta _0)\cap A_{\infty }$ are isolated, we can assume that $K(F,\beta _0)\cap A_{\infty }$ is isolated in $K(F,\beta _0)$, so that

$$\begin{aligned} \mathrm {dist}(x_k^m,K(F,\beta _0))\ge \varepsilon \;\, \text {for all}\;\, k\in {\mathbb {N}}, \end{aligned}$$

which furnishes the desired contradiction. $\square$

The next proposition is the same as Proposition 3.8 in the case of homological or cohomological admissible families.

Proposition 3.9

Let $d\ge 1$ be a fixed integer, R be an arbitrary ring, G be an abelian group, $F\in C^2(X,{\mathbb {R}}_+)$ as in Theorem 3.6, $B\subset X$ a compact subset, $\alpha _{*}\in H_d(X,B,R){\setminus }{\left\{ 0\right\} }$ and $\alpha ^{*}\in H^d(X,G){\setminus }{\left\{ 0\right\} }$ be non-trivial classes in relative homology and cohomology respectively, let $\underline{{\mathscr {A}}}(\alpha _{*})$ and $\overline{{\mathscr {A}}}(\alpha ^{*})$ be the corresponding d-dimensional homological and cohomological admissible families, and

$$\begin{aligned} \beta _0=\beta (F,\underline{{\mathscr {A}}}(\alpha _{*}))=\inf _{A\in \underline{{\mathscr {A}}}(\alpha _{*})}\sup F(A),\quad {\overline{\beta }}_0=\beta (F,\overline{{\mathscr {A}}}(\alpha ^{*}))=\inf _{A\in \overline{{\mathscr {A}}}(\alpha ^{*})}\sup F(A) \end{aligned}$$

be the associated width. Assume that $x_0\in K(F,\beta _0)$ (resp. $x_0\in K(F,{\overline{\beta }}_0)$) is a non-degenerate critical points of F at level $\beta _0$ (resp. ${\overline{\beta }}_0$) and that

$$\begin{aligned} \mathrm {Ind}_{F}(x_0)\ne d. \end{aligned}$$

(3.22)

Then for all small enough $\varepsilon >0$, there exists $\delta >0$ such that for all $A\in \underline{{\mathscr {A}}}(\alpha _{*})$ (resp. $\overline{{\mathscr {A}}}(\alpha ^{*})$), $\sup F(A)\le \beta _0+\delta$ (resp. $\sup F(A)\le {\overline{\beta }}_0+\delta$) implies that there exists $A'\in \underline{{\mathscr {A}}}(\alpha _{*})$ (resp. $\overline{{\mathscr {A}}}(\alpha ^{*})$) such that

$$\begin{aligned} \left\{ \begin{array}{ll} A{\setminus } U_{2\varepsilon }(x_0)=A'{\setminus } U_{2\varepsilon }(x_0)\\ A'\cap U_{\varepsilon }(x_0)=\varnothing \\ \sup F(A')\le \sup F(A). \end{array}\right. \end{aligned}$$

(3.23)

Proof

Let $x_0\in K(F,\beta _0)$ be a non-degenerate critical critical point, and let $r_1,r_2,\delta >0$ be given by Lemma 3.1 such that $0<\delta <r_2^2-4r_1^2$, and $A\in \underline{{\mathscr {A}}}(\alpha _{*})$ such that

$$\begin{aligned} \sup F(A)\le \beta _0+\delta . \end{aligned}$$

Then by definition, $\alpha \in \mathrm {Im}(\iota _{A,*})$, where $\iota _{A,*}:H_{d}(A,B)\rightarrow H_d(X,B)$ is the induced map in relative homology from the inclusion $\iota _{A}:A\rightarrow X$. We will now show that for all $1/2<\varepsilon <1$ close enough to 1, we have

$$\begin{aligned} A{\setminus } \mathrm {int}(C(\varepsilon \, r_1,\varepsilon \, r_2))\cup \left( C(r_1,0){\setminus } \mathrm {int}(C(\varepsilon r_1,0))\right) \in \underline{{\mathscr {A}}}(\alpha _{*}). \end{aligned}$$

We choose $r_1,r_2>0$ small enough such that C(s, t) is closed for all $s\le 2r_1$ and $t\le r_2$ by Theorem 2.3. Let

$$\begin{aligned} Y=A\cup C(r_1,0) \end{aligned}$$

and observe that

$$\begin{aligned} \mathrm {int}(Y{\setminus } \mathrm {int}(C(\varepsilon \,r_1,\varepsilon \,r_2)))\cup \mathrm {int}\left( Y\cap C(r_1,r_2)\right) =Y, \end{aligned}$$

and define for convenience of notations

$$\begin{aligned} A_{\varepsilon }(r_1,r_2)=C(r_1,r_2){\setminus } \mathrm {int}(C(\varepsilon \,r_1,\varepsilon \,r_2)). \end{aligned}$$

Therefore, we obtain the following Mayer–Vietoris commutative diagram

Now, we have by the proof of Lemma 3.1 that

$$\begin{aligned}&\Phi (C(r_1,r_2))=\Phi (C(r_1,0))=B_-(0,r_1)\simeq B^n(0,1)\subset {\mathbb {R}}^n,\nonumber \\&\Phi (X\cap {\left\{ x:F(x)\le \beta _0+\delta \right\} }\cap \partial C(r_1,r_2))=\Phi (\partial C(r_1,0))=\varphi ^{-1}(\partial B_-(0,r_1)) \end{aligned}$$

(3.24)

where $\simeq$ designs the equivalence up to homeomorphism, $\varphi$ is the Lipschitzian local homeomorphism given by the Morse lemma, and $B_-(0,r_1)$ is the closed ball of radius $r_1$ in the Hilbert space $T_{x_0}X$ corresponding to negative space of $\nabla ^2F(x)\in {\mathscr {L}}(T_xX)$. Let us show that for $0<\varepsilon <1$ large enough, we have

$$\begin{aligned} \Phi (Y\cap A_{\varepsilon }(r_1,r_2))\subset \varphi ^{-1}(B_-(0,r_1){\setminus } U_-(0,\varepsilon \,r_1))\simeq S^{n-1}. \end{aligned}$$

First, let us show that for $0<\varepsilon <1$ large enough, we have

$$\begin{aligned} \Phi (X\cap {\left\{ x: F(x)\le \beta _0+\delta \right\} }\cap A_{\varepsilon }(r_1,r_2))\subset \varphi ^{-1}(B_-(0,r_1){\setminus } U_-(0,\varepsilon r_1)). \end{aligned}$$

By contradiction, if there exists $x\in X\cap {\left\{ x: F(x)\le \beta _0+\delta \right\} }\cap A_{\varepsilon }(r_1,r_2)$ such that $\Phi (x)\in \varphi ^{-1}(U_-(0,\varepsilon r_1))$, as

$$\begin{aligned} \Phi (x)=\varphi ^{-1}\left( \varphi (x)_-\right) \end{aligned}$$

(3.25)

we must have $\Vert \varphi (x)_-\Vert <\varepsilon r_1$, and as $A_{\varepsilon }(r_1,r_2)=C(r_1,r_2){\setminus } \mathrm {int}(C(\varepsilon \,r_1,\varepsilon \,r_2))$, this implies that $\Vert \varphi (x)_+\Vert \ge \varepsilon r_2$, so that

$$\begin{aligned} F(x)=\beta _0+\Vert \varphi (x)_+\Vert ^2-\Vert \varphi (x)_-\Vert ^2=\beta _0+\Vert \varphi (x)_+\Vert ^2\ge \beta _0+\varepsilon ^2r_2^2-\varepsilon ^2r_1^2, \end{aligned}$$

and as $0<\delta <r_2^2-4r_1^2$, we obtain

$$\begin{aligned} \beta _0+\varepsilon ^2r_2^2-\varepsilon ^2r_1^2\le F(x)\le \beta _0+\delta <\beta _0+r_2^2-4r_1^2 \end{aligned}$$

and as $0<2r_1<r_2$, this yields to a contradiction if

$$\begin{aligned} 0<\sqrt{1-\frac{3r_1^2}{r_2^2-r_1^2}}\le \varepsilon <1. \end{aligned}$$

Furthermore, as we trivially have (by (3.25), valid for all $x\in C(r_1,r_2)$)

$$\begin{aligned} \Phi (C(r_1,0){\setminus } \mathrm {int}(C(\varepsilon \,r_1,0)))&=\Phi (\varphi ^{-1}(B_-(0,r_1){\setminus } U_-(0,\varepsilon r_1)))\\ {}&=\varphi ^{-1}(B_-(0,r_1){\setminus } U_-(0,\varepsilon r_1)), \end{aligned}$$

we obtain as $\partial B_-(0,r_1)$ is a retract by deformation of $B_-(0,r_1){\setminus } U_-(0,\varepsilon \,r_1)$, we obtain the identity

$$\begin{aligned} \Phi (Y\cap A_{\varepsilon }(r_1,r_2))=\varphi ^{-1}(B_-(0,r_1){\setminus } U_-(0,\varepsilon r_1))\simeq S^{n-1}. \end{aligned}$$

(3.26)

We also notice that the first equality in (3.24) implies that

$$\begin{aligned} \Phi (Y\cap C(r_1,r_2))=\varphi ^{-1}(B_-(0,r_1))\simeq B^n(0,1)\subset {\mathbb {R}}^n. \end{aligned}$$

(3.27)

Indeed, we have

$$\begin{aligned} Y\cap C(r_1,r_2)=(A\cap C(r_1,r_2))\cup C(r_1,0) \end{aligned}$$

and by (3.24), $\Phi (A\cap C(r_1,r_2))\subset \Phi (C(r_1,r_2))=\varphi ^{-1}(B_-(0,r_1))$ and $\Phi (C(r_1,0))=\varphi ^{-1}(B_-(0,r_1))$, which yields (3.27).

By (3.26) and (3.27), we obtain

$$\begin{aligned} H_d(\Phi (Y\cap C(r_1,r_2)))={\left\{ 0\right\} },\quad H_{d-1}(\Phi (Y\cap A_{\varepsilon }(r_1,r_2)))= {\left\{ 0\right\} } \end{aligned}$$

and we obtain the following exact sequence

and as $\mathrm {Im}(f)=\mathrm {Ker}(g)=H_d(\Phi (Y))$, we deduce that f is surjective. Now, as the map $\Phi :X\rightarrow X$ given by Lemma 3.1 is continuous on $X\cap {\left\{ x:F(x)\le \beta _0+\delta \right\} }$ and isotopic to the identity on $X\cap {\left\{ x:F(x)\le \beta _0+\delta \right\} }$ (which contains Y), we deduce that the $\Phi _{*}$ homomorphisms in the Mayer–Vietoris commutative diagram are isomorphism, so we have a surjection

In particular, the arrow ${\overline{h}}$ of the following we obtain a surjection

(3.28)

Now, if $B\subset A_1\subset A_2\subset X$ are any two subsets containing B, we write $\iota _{A_1,A_2}:A_1\hookrightarrow A_2$ the injection and $\iota _{A_1,A_2*}:H_d(A_1,B)\rightarrow H_d(A_2,B)$ the induced map in homology. As $A\subset A\cup C(r_1,0)=Y\subset X$, and $\iota _{A,X}=\iota _{Y,X}\circ \iota _{A,Y}$ we have

$$\begin{aligned} \iota _{A,X*}=\iota _{Y,X*}\circ \iota _{A,Y*}, \end{aligned}$$

and as $\alpha _{*}\in \mathrm {Im}(\iota _{A,X,*})\subset H_d(X,B)$, this implies that $\alpha _{*}\in \mathrm {Im}(\iota _{Y,X*})$, and by the surjectivity of the arrow in (3.28), we obtain

$$\begin{aligned} \alpha _{*}\in \mathrm {Im}\left( \iota _{Y{\setminus } \mathrm {int}(C(\varepsilon \,r_1,\varepsilon \,r_2)),X*}\right) , \end{aligned}$$

which by definition means that (notice that Y is compact)

$$\begin{aligned} Y{\setminus } \mathrm {int}(C(\varepsilon \,r_1,\varepsilon \,r_2))\in \underline{{\mathscr {A}}}(\alpha _{*}). \end{aligned}$$

Finally, for all $x\in C(r_1,0){\setminus } \mathrm {int}(C(\varepsilon r_1,0))$, we have

$$\begin{aligned} F(x)=\beta _0-\Vert \varphi (x)_-\Vert \le \beta _0-\varepsilon ^2r_1^2<\beta _0\le \sup F(A), \end{aligned}$$

so that

$$\begin{aligned} \sup F(Y{\setminus } \mathrm {int}(C(\varepsilon \, r_1,\varepsilon \,r_2)))\le \sup F(A). \end{aligned}$$

Using the exact same arguments of proof (with $A'=A{\setminus } \mathrm {int}(C(\varepsilon \,r_1,\varepsilon \,r_2))\cup C(r_1,0){\setminus } \mathrm {int}(C(\varepsilon \,r_1,0))$) thanks to the Mayer–Vietoris sequence for singular cohomology, we show the injectivity of the following arrow

$$\begin{aligned} H^d(A\cup C(r_1,0),G)\hookrightarrow H^d(A{\setminus } \mathrm {int}(C(\varepsilon \,r_1,\varepsilon \,r_2))\cup (C(r_1,0){\setminus } \mathrm {int}(C(\varepsilon r_1,0)),G) \end{aligned}$$

and this finishes the proof of the theorem. $\square$

Remark 3.10

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

Corollary 3.11

Under the hypotheses of Proposition 3.9, if $F\in C^2(X,{\mathbb {R}})$ and ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset \underline{{\mathscr {A}}}(\alpha _{*})$ (resp. ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset \overline{{\mathscr {A}}}(\alpha ^{*})$) such that

$$\begin{aligned} \sup F(A_k){\underset{k\rightarrow \infty }{\longrightarrow }} \beta (F,\underline{{\mathscr {A}}}(\alpha _{*})),\quad \left( \text {resp.}\;\, \sup F(A_k){\underset{k\rightarrow \infty }{\longrightarrow }} \beta (F,\overline{{\mathscr {A}}}(\alpha ^{*})) \right) . \end{aligned}$$

If $K(F,\beta (F,\underline{{\mathscr {A}}}(\alpha _{*})))$ (resp. $K(F,\beta (F,\overline{{\mathscr {A}}}(\alpha ^{*})))$) contains only non-degenerate critical points, there exists a sequence ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that $x_k\in A_k$ for all $k\in {\mathbb {N}}$ and $x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x\in K(F,\beta (F,\underline{{\mathscr {A}}}(\alpha _{*})))\cap A_{\infty }$ (resp. ${\overline{x}}\in K(F,\overline{{\mathscr {A}}}(\alpha ^{*}))\cap A_{\infty }$) such that

$$\begin{aligned} \mathrm {Ind}_{F}(x)=d,\quad (\text {resp.}\;\, \mathrm {Ind}_{F}({\overline{x}})=d). \end{aligned}$$

(3.29)

Proof

It is exactly the same as the proof of Theorem 3.6, using Proposition 3.9 instead of Proposition 3.8. $\square$

4 Proof of the main theorem

4.1 The entropy condition

Let X be a Finsler manifold and ${\left\{ F_{\sigma }\right\} }_{\sigma \in [0,1]}\subset C^1(X,{\mathbb {R}})$ such that for all $x\in X$, $\sigma \mapsto F_{\sigma }(x)$ is increasing. If ${\mathscr {A}}$ is any of the admissible families, we define for all $\sigma \in [0,1]$

$$\begin{aligned} \beta (\sigma )=\inf _{A\in {\mathscr {A}}}\sup F_\sigma (A)<\infty . \end{aligned}$$

(4.1)

As the function $\sigma \rightarrow \beta (\sigma )$ is increasing, it is differentiable almost everywhere (with respect to the 1-dimensional Lebesgue measure) and we have

$$\begin{aligned} \liminf _{\sigma \rightarrow 0}\beta '(\sigma )\left( \sigma \log \left( \frac{1}{\sigma }\right) \log \log \left( \frac{1}{\sigma }\right) \right) =0. \end{aligned}$$

Suppose by contradiction that this is not the case. Then there exists $\delta >0$ such that for $\sigma >0$ small enough

$$\begin{aligned} \beta (\sigma )-\beta (0)\ge \int _{0}^{\sigma }\beta '(t)dt\ge \delta \int _{0}^{\sigma }\frac{dt}{t\log \left( \frac{1}{t}\right) \log \log \left( \frac{1}{t}\right) }=\infty , \end{aligned}$$

which contradicts (4.1).

Definition 4.1

We say that $\beta$ satisfies the entropy condition at $\sigma >0$ if $\beta$ is differentiable at $\sigma$ and if

$$\begin{aligned} \beta '(\sigma )\le \frac{1}{\sigma \log \left( \frac{1}{\sigma }\right) \log \log \left( \frac{1}{\sigma }\right) }. \end{aligned}$$

In particular, there always exists a sequence of positive number ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}$ such that $\sigma _k{\underset{k\rightarrow \infty }{\longrightarrow }}0$ and $\beta$ verifies the entropy condition at $\sigma _k$.

4.2 The non-degenerate case

If X is a Finsler–Hilbert manifold and $F:X\rightarrow {\mathbb {R}}$ is a $C^2$ map, we let $\nabla F(x)\in T_xX$ and $\nabla ^2F(x)\in {\mathscr {L}}(T_xX)$ such that for all $x\in T_xX$, there holds

$$\begin{aligned}&DF(x)\cdot v=\langle \nabla F(x),v\rangle _x\\&D^2F(x)(v,w)=\langle \nabla ^2F(x)v,w\rangle \end{aligned}$$

The next result is a variant of [34], 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 $F,G\in C^2(X,{\mathbb {R}}_+)$, ${\mathscr {A}}$ an admissible min–max family, and define for $0\le \sigma <1$ the function $F_{\sigma }=F+\sigma ^2G$, and

$$\begin{aligned} \beta (\sigma )=\inf _{A\in {\mathscr {A}}}\sup F_{\sigma }(A)<\infty , \end{aligned}$$

and assume that the Energy bound (2) of Theorem 1.1 holds. Now suppose that $\beta$ is differentiable at $0<\sigma <1$ and satisfies the entropy condition, i.e.

$$\begin{aligned} \beta '(\sigma )\le \frac{1}{\log \left( \frac{1}{\sigma }\right) \log \log (\frac{1}{\sigma })}. \end{aligned}$$

(4.2)

Let ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}\subset (\sigma ,\infty )$ be such that $\sigma _k\rightarrow \sigma$, and ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset {\mathscr {A}}$ such that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }F_{\sigma _k}(A_k)\le \beta (\sigma _k)+(\sigma _k-\sigma ). \end{aligned}$$

Then for $0<\sigma \le e^{-\frac{4}{\beta (0)}}$, there exists a sequence ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that for all large enough $k\in {\mathbb {N}}$

$$\begin{aligned} (1)\;\,&\mathrm {dist}(x_k,A_k){\underset{k\rightarrow \infty }{\longrightarrow }}0\\ (2)\;\,&\beta (\sigma )-(\sigma _k-\sigma )\le F_{\sigma }(x_k)\le F_{\sigma _k}(x_k)\le \beta (\sigma _k)+(\sigma _k-\sigma )\\ (3)\;\,&\Vert DF_{\sigma }(x_k)\Vert {\underset{k\rightarrow \infty }{\longrightarrow }}0\\ (4)\;\,&\inf _{k\in {\mathbb {N}}}F(x_k)>0. \end{aligned}$$

In particular, if $F_{\sigma }$ verifies the Palais–Smale condition at $\beta (\sigma )$, there exists $x_{\sigma }\in K_{\beta (\sigma )}\cap A_{\infty }^{\sigma }$ (where $A_{\infty }^{\sigma }=A_{\infty }$ as defined in (3.19)) such that

$$\begin{aligned} \sigma ^2G(x_{\sigma })\le \frac{1}{\log \left( \frac{1}{\sigma }\right) \log \log \left( \frac{1}{\sigma }\right) }. \end{aligned}$$

Proof

Looking at Step 2 of the proof of Proposition 6.3 of [15] (it is written for geodesics, but the same proof work equally well in general, see [25]), we see that assuming by contradiction that for all for $k\ge 1$ large enough, we have for all $x\in X$ such that $\mathrm {dist}(x,A_k)\le \delta _k$ and

$$\begin{aligned} \beta (\sigma )-(\sigma _k-\sigma )\le F_{\sigma }(x)\le F_{\sigma _k}(x)\le \beta (\sigma _k)+(\sigma _k-\sigma ), \end{aligned}$$

(4.3)

then

$$\begin{aligned} \Vert DF_{\sigma _k}(x)\Vert \ge \delta _k>0 \end{aligned}$$

for some $\delta _k>0$ to be determined later, there exists a semi-flow ${\left\{ \varphi ^t_k\right\} }_{t\ge 0}:X\rightarrow X$ isotopic to the identity and preserving the boundary of ${\mathscr {A}}$ such that for all $0\le t\le \delta _k$ (as $\mathrm {dist}(x,\varphi ^t(x))\le t$ for all $t\ge 0$), and $x\in A_k$ such that (4.3) is satisfied, there holds

$$\begin{aligned} \frac{d}{dt}F_{\sigma }(\varphi ^t_k(x))\le -\delta _k. \end{aligned}$$

(4.4)

In particular, as $\varphi ^t_k(A)\in {\mathscr {A}}$, we have

$$\begin{aligned} \beta (\sigma )\le F_{\sigma }(\varphi _k^t(A_k)), \end{aligned}$$

so we deduce that for all $0\le t\le \delta _k$ by (4.4)

$$\begin{aligned} \beta (\sigma )&\le \sup F_{\sigma }(\varphi ^t(A_k))\le \sup F_{\sigma }(A)-t\delta _k\le \sup F_{\sigma _k}(A_k)-t\delta _k\nonumber \\&\le \beta (\sigma _k)+(\sigma _k-\sigma )-t\delta _k. \end{aligned}$$

(4.5)

Furthermore, as $\beta$ is differentiable at $\sigma$, we can assume that k is large enough such that

$$\begin{aligned} \beta (\sigma _k)\le \beta (\sigma )+(\beta '(\sigma )+1)(\sigma _k-\sigma ) \end{aligned}$$

(4.6)

so by (4.5) and (4.6), we have for $t=\delta _k$ and $\eta _k=\varphi _k^{t}:X\rightarrow X$

$$\begin{aligned} \sup F_{\sigma }(\eta _k(A))\le \beta (\sigma )+(\beta '(\sigma )+2)(\sigma _k-\sigma )-\delta _k^2. \end{aligned}$$

Therefore, choosing

$$\begin{aligned} \delta _k=\sqrt{2(\beta '(\sigma )+2)(\sigma _k-\sigma )}, \end{aligned}$$

we find that $\eta _k(A_k)\in {\mathscr {A}}$ so (recall that $\beta '\ge 0$)

$$\begin{aligned} \beta (\sigma )=\inf _{A\in {\mathscr {A}}}\sup F_{\sigma }(A)\le \sup F_{\sigma }(\eta _k(A_k))\le \beta (\sigma )-2(\sigma _k-\sigma )<\beta (\sigma ), \end{aligned}$$

a contradiction. Therefore, we see that there exists $x_k\in X$ such that

$$\begin{aligned}&(1)\;\,\mathrm {dist}(x_k,A_k)\le \delta _k=\sqrt{2(\beta '(\sigma )+2)(\sigma _k-\sigma )}{\underset{k\rightarrow \infty }{\longrightarrow }}0\nonumber \\&(2)\;\,\beta (\sigma )-(\sigma _k-\sigma )\le F_{\sigma }(x_k)\le F_{\sigma _k}(x_k)\le \beta (\sigma _k)+(\sigma _k-\sigma )\nonumber \\&(3)'\;\,\Vert DF_{\sigma _k}(x_k)\Vert \le \delta _k\nonumber \\&(4)\;\,F(x_k)\ge \frac{3}{4}\beta (0) \end{aligned}$$

(4.7)

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 to the Energy bound condition), and this concludes the proof (see [28] for the optimal hypotheses on $F_{\sigma }$ for this assertion to hold true). $\square$

Remark 4.3

If $Y\hookrightarrow X$ is a locally Lipschitzian embedded Hilbert–Finsler manifold, and ${\mathscr {A}}\subset {\mathscr {P}}(Y)$ is an admissible family (i.e. it is stable under locally Lipschitzian 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 $A\in {\mathscr {A}}$ will be preserved by the map $\varphi ^{t}_{\delta _k}$. Therefore, we obtain a sequence ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset Y$ such that (4.7) are satisfied with respect to the Finsler norm and distance on Y, and by the local Lipschitzian embedding, we also obtain $\mathrm {dist}_X(x_k,A_k){\underset{k\rightarrow \infty }{\longrightarrow }}0$, and $\Vert D F_{\sigma _k}\Vert _{X,x_k}{\underset{k\rightarrow \infty }{\longrightarrow }}0$. Using the Palais–Smale condition and the energy bound valid with respect to X, the end of the proof is identical.

Definition 4.4

Under the previous notations, we define the set of points satisfying the entropy condition as

$$\begin{aligned} {\mathscr {E}}(\sigma )=X\cap {\left\{ x:\sigma ^2G(x)\le \frac{1}{\log (\frac{1}{\sigma })\log \log (\frac{1}{\sigma })}\right\} }. \end{aligned}$$

Theorem 4.5

Let X be a $C^2$ Finsler manifold, $F,G\in C^2(X,{\mathbb {R}}_+)$, and define for all $\sigma \ge 0$ the function $F_{\sigma }=F+\sigma ^2G\in C^2(X,{\mathbb {R}}_+)$ and let ${\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, resp. $\widetilde{{\mathscr {A}}}$) be a d-dimensional admissible family (resp. a dual family, resp. a co-dual family). Assume that $F_{\sigma }$ satisfied the hypotheses of Theorem 1.1, and let ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset {\mathscr {A}}$ (resp. ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset {\mathscr {A}}^{*}$, resp. ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}\subset \widetilde{{\mathscr {A}}}$) be a min-maximising sequence such that

$$\begin{aligned} \sup F_{\sigma _k}(A_k)\le \beta (\sigma _k)+(\sigma _k-\sigma ) \end{aligned}$$

and assume that all critical points of $F_{\sigma }$ in $K_{\beta (\sigma )}\cap A_{\infty }^{\sigma }\cap {\mathscr {E}}(\sigma )$ are non-degenerate. Then for all $0<\sigma \le e^{-\frac{4}{\beta (0)}}$ such that $\beta$ satisfies the entropy condition at $\sigma$ (see (4.2) from Theorem 4.2), there exists $x_{\sigma }\in K_{\beta (\sigma )}\cap A_{\infty }^{\sigma }\cap {\mathscr {E}}(\sigma )$ (resp. $x_{\sigma }^{*}\in K_{\beta ^{*}(\sigma )}\cap A_{\infty }^{\sigma }\cap {\mathscr {E}}(\sigma )$, resp. ${\widetilde{x}}_{\sigma }\in K_{{\widetilde{\beta }}(\sigma )}\cap A_{\infty }^{\sigma }\cap {\mathscr {E}}(\sigma )$ ) such that

$$\begin{aligned} \left\{ \begin{array}{llllll} F_{\sigma }(x_{\sigma })=\beta (\sigma ),\quad &{}&{} \sigma ^2G(x_{\sigma })\le \frac{1}{\log (\frac{1}{\sigma })\log \log \left( \frac{1}{\sigma }\right) },\quad &{}&{} \text {and}\;\; \mathrm {Ind}_{F_{\sigma }}(x_{\sigma })\le d.\\ F_{\sigma }(x_{\sigma }^{*})=\beta ^{*}(\sigma ),\quad &{}&{} \sigma ^2G(x_{\sigma }^{*})\le \frac{1}{\log \left( \frac{1}{\sigma }\right) \log \log \left( \frac{1}{\sigma }\right) },\quad &{}&{} \text {and}\;\, \mathrm {Ind}_{F_{\sigma }}(x_{\sigma }^{*})\ge d\\ F_{\sigma }({\widetilde{x}}_{\sigma })={\widetilde{\beta }}(\sigma ),\quad &{}&{} \sigma ^2G({\widetilde{x}}_{\sigma })\le \frac{1}{\log \left( \frac{1}{\sigma }\right) \log \log \left( \frac{1}{\sigma }\right) },\quad &{}&{} \text {and}\;\, \mathrm {Ind}_{F_{\sigma }}({\widetilde{x}}_{\sigma })=d \end{array}\right. \end{aligned}$$

(4.8)

Remark 4.6

Likewise, the proof would work equally well for homotopical and cohomotopical families, by Proposition 3.9.

Proof

Theorem 4.5 is an adaptation of Theorem 3.6 of Lazer–Solimini for the viscosity method. As Proposition 3.8, the Palais–Smale condition and the non-degeneracy gives Theorem 3.6, Theorem 4.5 follows by Proposition 3.8, Theorem 4.2 and the non-degeneracy assumption.

We give the proof in the special case where X is $C^3$ and $F,G\in C^3(X,{\mathbb {R}})$, 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.

Let $K= K_{\beta (\sigma )}\cap A_{\infty }^{\sigma }\cap {\mathscr {E}}(\sigma )$ (notice that $K\ne \varnothing$ thanks to Theorem 4.2). As the critical points in K are non-degenerate, K is compact and consists of finitely many points ${\left\{ {\overline{x}}_0,{\overline{x}}_1,\ldots ,{\overline{x}}_m\right\} }\subset K_{\beta (\sigma )}$. We cannot apply the previous lemma on $F_{\sigma }$ as the main lemma only work with $F_{\sigma _k}$.

Part 1 In the first part of the proof, we will show that we apply Proposition 3.8 to $F_{\sigma _k}$ with a $\delta >0$ and $\varepsilon >0$ independent of k. This will permit to get a contradiction using Theorem 4.2.

First, by the Palais–Smale condition for $F_{\sigma }$ and as the critical points are isolated, we deduce that there exists $\delta >0$ such that $B_{2\delta }(x_i)\cap B_{2\delta }(x_j)=\varnothing$ for all $i\ne j$ and

$$\begin{aligned} \Vert DF_{\sigma }(x)\Vert \ge \delta \;\, \text {for all}\;\, x\in U_{2\delta }(K){\setminus } U_{\delta }(K). \end{aligned}$$

Also notice that thanks to the proof of Theorem 4.2, for all ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} \Vert DF_{\sigma _k}(x_k)\Vert {\underset{k\rightarrow \infty }{\longrightarrow }}0\\ |F_{\sigma _k}(x_k)-\beta (\sigma _k)|{\underset{k\rightarrow \infty }{\longrightarrow }}0 \end{array}\right. \end{aligned}$$

(4.9)

then

$$\begin{aligned} \left\{ \begin{array}{ll} \Vert DF_{\sigma }(x_k)\Vert {\underset{k\rightarrow \infty }{\longrightarrow }}0\\ |DF_{\sigma }(x_k)-\beta (\sigma )|{\underset{k\rightarrow \infty }{\longrightarrow }}0 \end{array}\right. \end{aligned}$$

so up to a subsequence, we have thanks to the Palais–Smale condition for $F_{\sigma }$ at level $\beta (\sigma )$ that $x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x\in K_{\beta (\sigma )}$. In particular, if ${\left\{ x_k\right\} }\subset X$ verifies (4.9), then we can assume up to some relabelling that that for all $k\in {\mathbb {N}}$ large enough $x_k\in N_{\delta }({\overline{x}}_0)$. Now, looking at the proof of Morse Lemma by Palais ([19]) which only works for $C^3$ functions, we see that the diffeomorphism $\varphi$ around a critical point ${\overline{x}}_i$ such that

$$\begin{aligned} F_{\sigma }(\varphi _{{\overline{x}}_0}(x))=\beta (\sigma )+\Vert x_+\Vert ^2-\Vert x_-\Vert ^2 \end{aligned}$$

is defined by

$$\begin{aligned} \varphi _{{\overline{x}}_0}(x)=\sqrt{A_{{\overline{x}}_0}({\overline{x}}_0)^{-1}A_{{x}_0}(x)}x, \end{aligned}$$

where for all $v,w\in H$, we have by Taylor expansion for some map $A_{{\overline{x}}_0}:B({\overline{x}}_0,\delta )\rightarrow {\mathscr {L}}(H)$ with values into self-adjoint continuous operators

$$\begin{aligned}&F_{\sigma }(x)=\beta (\sigma )+\langle A_{{\overline{x}}_0}(x)x,x\rangle \\&D^2F_{\sigma }({\overline{x}}_0)(v,w)=2\langle A_{{\overline{x}}_0}({\overline{x}}_0)v,w\rangle \end{aligned}$$

Now, notice if $B_{{\overline{x}}_0}(x)=A_{{\overline{x}}_0}({\overline{x}}_0)^{-1}A_{{\overline{x}}_0}(x)$ that $B_{{\overline{x}}_0}({\overline{x}}_0)=\mathrm {Id}_{H}$ and as for $\Vert h\Vert <1$, $\sqrt{\mathrm {Id}+h}$ is well defined by the absolutely convergent series

$$\begin{aligned} \sqrt{\mathrm {Id}+h}=\sum _{n=0}^{\infty }\left( {\begin{array}{c}\frac{1}{2}\\ n\end{array}}\right) h^n, \end{aligned}$$

we deduce that for some $\delta _0>0$ small enough and depending only on $A_{{\overline{x}}_0}$, namely such that for all $x\in B(x,\delta )$

$$\begin{aligned} \Vert x-B_{{\overline{x}}_0}(x)\Vert <\frac{1}{2}\;\,\text {for all}\;\, x\in B({\overline{x}}_0,\delta )\quad \end{aligned}$$

(4.10)

that $\varphi (x)$ is well-defined on $B({\overline{x}}_0,\delta )$ and $C^1$. Therefore, thanks to the local inversion theorem, up to diminishing $\delta$, we can assume that $\varphi$ is a diffeomorphism from $B({\overline{x}}_0,\delta )$ onto its image (here, $\delta$ depends only on $A_{{\overline{x}}_0}$).

Now, let $x_k\in K_{\beta (\sigma _k)}$ be a critical point of $F_{\sigma _k}$ and $A_{x_k}$ such that

$$\begin{aligned}&F_{\sigma _k}(x)=\beta (\sigma _k)+\langle A_{x_k}(x)x,x\rangle \\&DF^2_{\sigma _k}(x)=2\langle A_{x_k}(x_k)x,x\rangle \end{aligned}$$

such that $x_k{\underset{k\rightarrow \infty }{\longrightarrow }}{\overline{x}}_0$. Thanks to the strong convergence, we deduce that for k large enough, $A_{x_k}(x_k)$ is an invertible operator so we can define for k large enough $B_{x_k}(x)=A_{x_k}(x_k)^{-1}A_{x_k}(x)$. Now, taking k large enough such that $B(x_k,\frac{\delta }{2})\subset B({\overline{x}}_0,\delta )$, we see by the strong convergence of $x_k\rightarrow {\overline{x}}_0$ that

$$\begin{aligned} \Vert B_{x_k}-B_{{\overline{x}}_0}\Vert _{B(x_k,\frac{\delta }{2})}{\underset{k\rightarrow \infty }{\longrightarrow }}0. \end{aligned}$$

In particular, if k is large enough such that

$$\begin{aligned} \Vert B(x_k)(x)-B_{{\overline{x}}_0}(x)\Vert \le \frac{1}{2}\;\, \text {for all}\;\, x\in B\left( x_k,\frac{\delta }{2}\right) , \end{aligned}$$

we deduce by (4.10) that

$$\begin{aligned} \Vert x-B_{x_k}(x)\Vert <1\;\, \text {for all}\;\, x\in B\left( x_k,\frac{\delta }{2}\right) . \end{aligned}$$

In particular, we can define $\varphi _{x_k}(x)=\sqrt{B_{x_k}(x)}x$ for all $x\in B(x,\frac{\delta }{2})$, and we see that in particular $d\varphi _k(x_k)=\mathrm {Id}$. Now, as

$$\begin{aligned} \Vert \varphi _{x_k}-\varphi _{{\overline{x}}_0}\Vert _{C^1(B(x_k,\frac{\delta }{2}))}{\underset{k\rightarrow \infty }{\longrightarrow }}0, \end{aligned}$$

and as the neighbourhood around which $\varphi _{x_k}$ is invertible depends only on the local behaviour of its derivative around $x_k$ and as $\varphi _{{\overline{x}}_0}$ is invertible in $B({\overline{x}}_0,\delta )$, we deduce that for k large enough, $\varphi _k$ is invertible on $B(x_k,\frac{\delta }{4})$, so the Morse lemma implies that

$$\begin{aligned} F_{\sigma _k}(\varphi _k(x))=\beta (\sigma _k)+\Vert x_+^k\Vert ^2-\Vert x^k_-\Vert ^2\;\,\text {for all}\;\, x\in B\left( x_k,\frac{\delta }{4}\right) \end{aligned}$$

In particular, $F_{\sigma _k}$ has only one critical point on $B({\overline{x}}_0,\frac{\delta }{8})\subset B(x_k,\frac{\delta }{4})$ for k large enough. Therefore, we can apply the Proposition 3.8 to $F_{\sigma _k}$ with $\delta >0$ and $\varepsilon >0$ independent of k.

Part 2 End of the proof.

As $K={\left\{ {\overline{x}}_0,{\overline{x}}_1,\ldots ,{\overline{x}}_m\right\} }$ is finite, we saw that for all k sufficiently large, $F_{\sigma _k}$ has at most one critical point in $B(x_i,\frac{\delta }{8})$. Let us denote by $K_{\beta (\sigma _k)}\cap U_{\delta /8}(K)={\left\{ x_0^k,x_1^k,\ldots ,x_{m_k}^k\right\} }$ where $m_k\le m$ the critical points of $F_{\sigma _k}$ at level $\beta (\sigma _k)$. Thanks to Proposition 3.8 and the first part of the proof, there exists some $\delta >0$ independent of k such that for all ${A}\in {\mathscr {A}}$ such that $\sup F_{\sigma _k}(A)\le \beta (\sigma )+\delta$, then for all $1\le i\le m_{k}$, there exists $A'_i\in {\mathscr {A}}$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} A{\setminus } U_{2\varepsilon }(x_i^k)=A_i'{\setminus } U_{2\varepsilon }(x_i^k)\\ A_i'\cap U_{\varepsilon }(x_i^k)=\varnothing \\ \sup F_{\sigma _k}(A_i')\le \sup F_{\sigma _k}(A). \end{array}\right. \end{aligned}$$

(4.11)

Furthermore, as the $x_i^k$ are uniformly isolated independently of k, taking $\varepsilon >0$ small enough, we can assume that

$$\begin{aligned}&U_{2\varepsilon }(x_i^k)\cap U_{2\varepsilon }(x_j^k)=\varnothing \quad \text {for all }\;\,1\le i\ne j\le m_k . \end{aligned}$$

(4.12)

and that if $x_i^k{\underset{k\rightarrow \infty }{\longrightarrow }}x_j\in K$ (for some $j\in {\left\{ 1,\ldots ,m\right\} }$) that k is large enough such that

$$\begin{aligned} U_{\frac{\varepsilon }{2}}(x_j)\subset U_{\varepsilon }(x_i^k). \end{aligned}$$

We also remark that

$$\begin{aligned} U_{\frac{\varepsilon }{2}}(K)=\bigcup _{i=1}^nU_{\frac{\varepsilon }{2}}(x_i) \end{aligned}$$

(4.13)

is an open neighbourhood of $K=K_{\beta (\sigma )}\cap A_{\infty }^{\sigma }\cap {\mathscr {E}}(\sigma )$. Now, let ${\left\{ \sigma _k\right\} }\subset (\sigma ,\infty )$ such that $\sigma _k{\underset{k\rightarrow \infty }{\longrightarrow }}\sigma$ and ${\left\{ A_k\right\} }_{k\in {\mathbb {N}}}$ such that

$$\begin{aligned} \sup F_{\sigma _k}(A_k)\le \beta (\sigma _k)+(\sigma _k-\sigma ){\underset{k\rightarrow '\infty }{\longrightarrow }}\beta (\sigma ). \end{aligned}$$

In particular, there exists $k_0\in {\mathbb {N}}$ such that for all $k\ge k_0$, there holds

$$\begin{aligned} \sup F_{\sigma _k}(A_k)\le \beta (\sigma )+\delta . \end{aligned}$$

We can also assume as K is isolated in $K_{\beta (\sigma )}\cap {\mathscr {E}}(\sigma )$ and thanks to the first part of the proof that $\varepsilon >0$ is small enough such that

$$\begin{aligned} A_k\cap U_{\frac{\varepsilon }{2}}\left( (K_{\beta (\sigma )}\cap {\mathscr {E}}(\sigma )){\setminus } K\right) =\varnothing \;\, \text {for all}\;\, k \;\, \text {large enough}. \end{aligned}$$

(4.14)

Now, define by induction a finite sequence (recall that $m_k\le m$) $A^0_k,A_k^1,\ldots ,A_k^{m_k}\in {\mathscr {A}}$ by $A_k^0=A_k$, $A_k^1=(A_k^0)_1'=(A_k)_0'$,

$$\begin{aligned} A_k^j=(A_k^{j-1})'_j \end{aligned}$$

using the notation of (4.11). We see in particular that by (4.11) and (4.12)

$$\begin{aligned} A_k^j\cap U_{\varepsilon }(x_i^k)=\varnothing \;\, \text {for all}\;\, 1\le i\le j\le m_k. \end{aligned}$$

(4.15)

Furthermore, as for all $1\le j\le m_k$, we have by (4.11)

$$\begin{aligned} \sup F_{\sigma _k}(A_k^j)\le \sup F_{\sigma _k}(A_k^{j-1}) \end{aligned}$$

(4.16)

so by combining (4.14), (4.13) with (4.15) and (4.16), we deduce that for all $k\ge k_0$, we have

$$\begin{aligned}&A^{m_k}_k\cap U_{\frac{\varepsilon }{2}}(K_{\beta (\sigma )}\cap {\mathscr {E}}(\sigma ))=\varnothing \end{aligned}$$

(4.17)

$$\begin{aligned}&\beta (\sigma )\le \sup F_{\sigma _k}(A_k^{m_k})\le \sup F_{\sigma _k}(A_k)\le \beta (\sigma _k)+(\sigma _k-\sigma ){\underset{k\rightarrow \infty }{\longrightarrow }}\beta (\sigma ). \end{aligned}$$

(4.18)

By Theorem 4.2 there exists a sequence ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} (1)\;\,&{}\mathrm {dist}(x_k,A_k^{m_k}){\underset{k\rightarrow \infty }{\longrightarrow }}0\\ (2)\;\,&{}\beta (\sigma )-(\sigma _k-\sigma )\le F_{\sigma }(x_k)\le F_{\sigma _k}(x_k)\le \beta (\sigma _k)+(\sigma _k-\sigma )\\ (3)\;\,&{}\Vert DF_{\sigma }(x_k)\Vert {\underset{k\rightarrow \infty }{\longrightarrow }}0\\ (4)\;\,&{}\inf _{k\in {\mathbb {N}}}F(x_k)>0. \end{array}\right. \end{aligned}$$

(4.19)

Therefore, by the Palais–Smale condition at level $\beta (\sigma )$ and (4.19), up to a subsequence we have $x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x_{\infty }\in K_{\beta (\sigma )}\cap {\mathscr {E}}(\sigma )$. However, we have for all k large enough by (4.17) and as $\mathrm {dist}(x_k,A_k^{m_k}){\underset{k\rightarrow \infty }{\longrightarrow }}0$

$$\begin{aligned} \mathrm {dist}(A_k^{m_k},K_{\beta (\sigma )}\cap {\mathscr {E}}(\sigma ))\ge \frac{\varepsilon }{4}, \end{aligned}$$

and this contradicts the fact that $x_{\infty }\in K_{\beta (\sigma )}\cap {\mathscr {E}}(\sigma )$. This concludes the proof of the theorem. $\square$

4.3 Marino–Prodi perturbation method and the degenerate case

Let us recall the main theorem here.

Theorem 4.7

Let $(X,\Vert \,\cdot \,\Vert _X)$ be a $C^2$ Finsler manifold modelled on a Banach space E, and $Y\hookrightarrow X$ be a $C^2$ Finsler–Hilbert manifold modelled on a Hilbert space H which we suppose locally Lipschitz embedded in X, and let $F,G\in C^2(X,{\mathbb {R}}_+)$ be two fixed functions. Define for all $\sigma >0$, $F_\sigma =F+\sigma ^2G\in C^2(X,{\mathbb {R}}_+)$ and suppose that the following conditions hold.

(1):

Palais–Smale condition: For all $\sigma >0$, the function $F_{\sigma }:X\rightarrow Y$ satisfies the Palais–Smale condition at all positive level $c>0$.

(2):

Energy bound: The following energy bound condition holds : for all $\sigma >0$ and for all ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that

$$\begin{aligned} \sup _{k\in {\mathbb {N}}}F_{\sigma }(x_k)<\infty , \end{aligned}$$

we have

$$\begin{aligned} \sup _{k\in {\mathbb {N}}}\Vert \nabla G(x_k)\Vert <\infty . \end{aligned}$$

(3):

Fredholm property: For all $\sigma >0$ and for all $x\in K(F_{\sigma })$, we have $x\in Y$, and the second derivative $D^2F_{\sigma }(x):T_xX\rightarrow T_x^{*}X$ restrict on the Hilbert space $T_xY$ such that the linear map $\nabla ^2F_{\sigma }(x)\in {\mathscr {L}}(T_yY)$ defined by

$$\begin{aligned} D^2F_{\sigma }(x)(v,v)=\langle \nabla ^2F_{\sigma }(x)v,v\rangle _{Y,x},\quad \text {for all}\;\, v\in T_xY, \end{aligned}$$

is a Fredholm operator, and the embedding $T_xY\hookrightarrow T_xX$ is dense for the Finsler norm $\Vert \,\cdot \,\Vert _{X,x}$.

Now, let ${\mathscr {A}}$ (resp. ${\mathscr {A}}^{*}$, resp. $\overline{{\mathscr {A}}}$, resp. $\underline{{\mathscr {A}}}(\alpha _{*})$, resp. $\overline{{\mathscr {A}}}(\alpha ^{*})$, where the last two families depend respectively on a homology class $\alpha _{*}\in H_d(Y,B)$-where $B\subset Y$ is a fixed compact subset-and a cohomology class $\alpha ^{*}\in H^d (Y)$) be a d-dimensional admissible family of Y (resp. a d-dimension dual family to ${\mathscr {A}}$, resp. a d-dimensional co-dual family to ${\mathscr {A}}^{*}$, resp. a d-dimensional homological family, resp. a d-dimensional co-homological family) with boundary ${\left\{ C_i\right\} }_{i\in I}\subset Y$. Define for all $\sigma >0$

$$\begin{aligned} \begin{array}{llllll} \beta (\sigma )=\inf _{A\in {\mathscr {A}}}\sup F_\sigma (A)<\infty ,\quad &{}&{} \beta ^{*}(\sigma )=\inf _{A\in {\mathscr {A}}^{*}}\sup F_{\sigma }(A), \quad &{}&{}{\widetilde{\beta }}(\sigma )=\inf _{A\in \widetilde{{\mathscr {A}}}}\sup F_{\sigma }(A)\\ {\overline{\beta }}(\sigma )=\inf _{A\in \underline{{\mathscr {A}}}(\alpha _{*})}\sup F_{\sigma }(A),\quad &{}&{} {\underline{\beta }}(\sigma )=\inf _{A\in \overline{{\mathscr {A}}}(\alpha ^{*})}\sup F_{\sigma }(A). \end{array} \end{aligned}$$

Assuming that the min–max value is non-trivial, i.e.

• (4) $\mathbf{Non-trivialilty: }$ $\beta _0=\inf _{A\in {\mathscr {A}}}\sup F(A)>\sup _{i\in I} \sup F(C_i)={\widehat{\beta }}_0$,

there exists a sequence ${\left\{ \sigma _k\right\} }_{k\in {\mathbb {N}}}\subset (0,\infty )$ such that $\sigma _k{\underset{k\rightarrow \infty }{\longrightarrow }}0$, and for all $k\in {\mathbb {N}}$, there exists a critical point $x_k\in K(F_{\sigma _k})\in {\mathscr {E}}(\sigma _k)$ (resp. $x_k^{*},{\widetilde{x}}_k,{\underline{x}}_k,{\overline{x}}_{k}\in K(F_{\sigma _k})\cap {\mathscr {E}}(\sigma _k)$) of $F_{\sigma _k}$ satisfying the entropy condition (1.3) (where we recall that $x_k\in Y$ by the condition (3)) satisfying the entropy condition (1.3) (see Definition 4.4) and such that respectively

$$\begin{aligned} \left\{ \begin{array}{llll} F_{\sigma _k}(x_k)=\beta (\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}(x_k)\le d\\ F_{\sigma _k}(x_k^{*})=\beta ^{*}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}(x_{k})\ge d \\ F_{\sigma _k}({\widetilde{x}}_k)={\widetilde{\beta }}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}({\widetilde{x}}_k)\le d\le \mathrm {Ind}_{F_{\sigma _k}}({\widetilde{x}}_k)+\mathrm {Null}_{F_{\sigma _k}}({\widetilde{x}}_k) \\ F_{\sigma _k}({\overline{x}}_k)={\overline{\beta }}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}({\overline{x}}_k)\le d\le \mathrm {Ind}_{F_{\sigma _k}}({\overline{x}}_k)+\mathrm {Null}_{F_{\sigma _k}}({\overline{x}}_k) \\ F_{\sigma _k}({\underline{x}}_k)={\underline{\beta }}(\sigma _k),\quad &{}&{} \mathrm {Ind}_{F_{\sigma _k}}({\underline{x}}_k)\le d\le \mathrm {Ind}_{F_{\sigma _k}}({\underline{x}}_k)+\mathrm {Null}_{F_{\sigma _k}}({\underline{x}}_k). \end{array}\right. \end{aligned}$$

Proof

As we have mentioned already, we can assume that X is a Finsler–Hilbert manifold modelled on a Hilbert space H. Take $\sigma >0$ such that $\beta$ satisfies the entropy condition at $\sigma$. If $F_{\sigma }$ has only non-degenerate critical points in $K_{\beta (\sigma )}\cap {\mathscr {E}}(\sigma )$, then we are done. $\square$

Lemma 4.8

Let ${\left\{ a_j\right\} }_{j\in {\mathbb {N}}}\subset [0,\infty )$ and ${\left\{ b_j\right\} }_{j\in {\mathbb {N}}}\subset (0,\infty )$ be two sequences such that

$$\begin{aligned} \sum _{j\in {\mathbb {N}}}^{}a_j<\infty \quad \text {and}\quad \sum _{j\in {\mathbb {N}}}^{}b_j=\infty . \end{aligned}$$

Then there holds

$$\begin{aligned} \liminf _{j\rightarrow \infty }\frac{a_j}{b_j}=0. \end{aligned}$$

Proof

By contradiction, let $\delta >0$ such that

$$\begin{aligned} \liminf \limits _{j\rightarrow \infty }\frac{a_j}{b_j}=\delta . \end{aligned}$$

Then there exists $J\in {\mathbb {N}}$ such that for all $j\ge J$,

$$\begin{aligned} \frac{a_j}{b_j}\ge \frac{\delta }{2}, \end{aligned}$$

so that for all $j\ge J$, there holds $\delta \, b_j\le 2a_j$. Therefore, we obtain

$$\begin{aligned} \sum _{j\ge J}b_j\le \frac{2}{\delta }\sum _{j\ge J}^{}a_j<\infty , \end{aligned}$$

contradicting the divergence of $\sum b_j$. $\square$

Let ${\left\{ a_j\right\} }_{j\in {\mathbb {N}}}\subset (0,1)$ be a strictly decreasing sequence converging to zero. Then there holds as $\beta$ is increasing for all $j\in {\mathbb {N}}$ by [6], 2.6.19 (2)

$$\begin{aligned} \int _{a_{j+1}}^{a_j}\beta '(\sigma )d\sigma \le \beta (a_j)-\beta (a_{j+1}) \end{aligned}$$

and we notice that

$$\begin{aligned} \sum _{j=0}^{n}\left( \beta (a_j)-\beta (a_{j+1})\right) = \beta (a_0)-\beta (a_{n+1}){\underset{n\rightarrow \infty }{\longrightarrow }}\beta (a_0)-\beta (0)<\infty . \end{aligned}$$

This implies that

$$\begin{aligned} \sum _{j\in {\mathbb {N}}}^{}\left( \beta (a_j)-\beta (a_{j+1})\right) <\infty . \end{aligned}$$

Therefore, if $b={\left\{ b_j\right\} }_{j\in {\mathbb {N}}}$ is a the general term of a divergent series with positive terms, there exists by Lemma 4.8 a subsequence ${\left\{ j_l\right\} }_{l\in {\mathbb {N}}}$ such that for all $l\in {\mathbb {N}}$, there holds

$$\begin{aligned} \beta (a_{j_l})-\beta (a_{j_l+1})\le b_{j_l}. \end{aligned}$$

(4.20)

Now, for convenience of notation, as we do not use any properties related to the convergence of the series of general term ${\left\{ b_{j_l}\right\} }_{l\in {\mathbb {N}}}$, we will assume that (4.20) holds for all $j\in {\mathbb {N}}$. Now, we want to find such sequence ${\left\{ a_j\right\} }_{j\in {\mathbb {N}}}$ and ${\left\{ b_j\right\} }_{j\in {\mathbb {N}}}$ such that

$$\begin{aligned} (a_j-a_{j+1})^{-1}b_j\le \frac{1}{a_j\log \left( \frac{1}{a_j}\right) \log \log \left( \frac{1}{a_j}\right) \log \log \log \left( \frac{1}{a_j}\right) } \end{aligned}$$

Take $a_j=\dfrac{1}{j}$, we have $a_j-a_{j+1}=\dfrac{1}{j(j+1)}$, so the condition becomes for $j\ge 4\cdot 10^{6}>e^{e^e}$

$$\begin{aligned} b_j\le \frac{1}{(j+1)\log (j)\log \log (j)\log \log \log (j)}, \end{aligned}$$

(4.21)

and the series whose general term is the right-hand side of (4.21) diverges so we define ${\left\{ b_j\right\} }_{j\in {\mathbb {N}}}\subset (0,\infty )^{{\mathbb {N}}}$ such that for all $j\ge J\ge 4\cdot 10^{6}>e^{e^e}$

$$\begin{aligned} b_j=\frac{1}{(j+1)\log (j)\log \log (j)\log \log \log (j)}. \end{aligned}$$

Now, for all $j\ge J$, let $I_j=[a_{j+1},a_j]$ and

$$\begin{aligned} A_j=I_j\cap {\left\{ \sigma :\beta '(\sigma )\le \frac{1}{a_j\log \left( \frac{1}{a_j}\right) \log \log \left( \frac{1}{a_j}\right) }\le \frac{1}{\sigma \log \left( \frac{1}{\sigma }\right) \log \log \left( \frac{1}{\sigma }\right) }\right\} }, \end{aligned}$$

and define $\delta _j$ for $j\ge J$ by

$$\begin{aligned} \delta _j=\frac{1}{\log \log \log (j)}{\underset{j\rightarrow \infty }{\longrightarrow }}0. \end{aligned}$$

Then for all $j\ge J$, there holds by (4.20)

$$\begin{aligned} \int _{a_{j+1}}^{a_j}\beta '(\sigma )d\sigma \le \frac{\delta _j(a_j-a_{j+1})}{a_j\log (\frac{1}{a_j})\log \log (\frac{1}{a_j})} \end{aligned}$$

so that

$$\begin{aligned} \frac{{\mathscr {L}}^1(I_j{\setminus } A_j)}{a_j\log \left( \frac{1}{a_j}\right) \log \log \left( \frac{1}{a_j}\right) }\le \int _{I_j{\setminus } A_j}\beta '(\sigma )d\sigma \le \int _{I_j}\beta '(\sigma )d\sigma \le \frac{\delta _j{\mathscr {L}}^1(I_j)}{a_j\log \left( \frac{1}{a_j}\right) \log \log \left( \frac{1}{a_j}\right) } \end{aligned}$$

so that

$$\begin{aligned} \frac{{\mathscr {L}}^1(I_j{\setminus } A_j)}{{\mathscr {L}}^1(I_j)}\le \delta _j{\underset{j\rightarrow \infty }{\longrightarrow }}0. \end{aligned}$$

(4.22)

Therefore, we obtain for all $j\ge J$ some element $\sigma _j\in (a_{j+1},a_j)$ such that

$$\begin{aligned} \beta '(\sigma _j)\le \frac{\beta (a_j)-\beta (a_{j+1})}{a_j-a_{j+1}}\le \frac{1}{a_j\log (\frac{1}{a_j})\log \log (\frac{1}{a_j})}\le \frac{1}{\sigma _j\log (\frac{1}{\sigma _j})\log \log (\frac{1}{\sigma _j})}. \end{aligned}$$

Now, for all $\sigma \in (0,1)$, as $K(F_{\sigma })$ is compact, we let $\varphi _{\sigma }$ be the cut-off function given by Proposition 2.20 and let $\varepsilon (\sigma )>0$ such that for all $\Vert y\Vert < \varepsilon (\sigma )$ small enough such that by Proposition 2.16, the map

$$\begin{aligned} F_{\sigma ,y}=F_{\sigma }+\varphi _{\sigma }\langle y,\,\cdot \,\rangle \end{aligned}$$

(4.23)

is proper on $N_{2\delta }(K)$. Now, fix some $C>0$.

Claim 1 There exists $\delta (C,\sigma )>0$ (taken such that $\delta (C,\sigma )<\varepsilon (\sigma )$) such that for all $|\tau -\sigma |<\delta (\sigma )$, the map

$$\begin{aligned} F^{\tau }_{\sigma ,y}=F_{\sigma ,y}+(\tau ^2-\sigma ^2)G=F_{\tau }+\varphi _{\sigma }\langle y,\,\cdot \,\rangle \end{aligned}$$

is such that

$$\begin{aligned} K(F^{\tau }_{\sigma ,y})\cap {\left\{ x:F_{\sigma ,y}^{\tau }(x)\le C\right\} }\subset K(F_{\sigma })^{\delta }, \end{aligned}$$

(4.24)

and that for all y such that $F_{\sigma ,y}$ is non-degenerate (such y form a non-meager subset by Sard–Smale theorem and Proposition 2.20), the map $F^{\tau }_{\sigma ,y}$ has only non-degenerate critical points below the critical level $C>0$ (in practise we can just take $C=\beta (1)+1$, but $C=\beta (\sigma _0)+\eta$ for some $\sigma _0,\eta >0$ would work equally well).

First, observe that $F_{\sigma ,y}$ has no critical points in $X{\setminus } K(F_{\sigma })^{2\delta }$, as $F_{\sigma ,y}=F_{\sigma }$ in $X{\setminus } K(F_{\sigma })^{2\delta }$. Now, by contradiction, assume that there exists ${\left\{ \tau _k\right\} }_{k\in {\mathbb {N}}}$ such that $\tau _k\rightarrow \sigma$ and a sequence of critical points ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ (i.e. such that $x_k\in K(F_{\sigma ,y}^{\tau _k})\cap {\left\{ x:F_{\sigma }^{\tau }(x)\le C\right\} }$ for all $k\in {\mathbb {N}}$) and

$$\begin{aligned} \mathrm {dist}(x_k,K(F_{\sigma }))\ge \delta . \end{aligned}$$

Then, by the same proof mutadis mutandis of (6.9) of Proposition 6.3 in [15], we have thanks to the condition (2) on the energy bound that

$$\begin{aligned} \Vert \nabla F_{\sigma ,y}^{\tau _k}(x_k)-\nabla F_{\sigma ,y}(x_k)\Vert =(\tau _k^2-\sigma ^2)\Vert \nabla G(x_k)\Vert {\underset{k\rightarrow \infty }{\longrightarrow }}0 \end{aligned}$$

The proof is immediate here thanks to the energy bound, and in general, using (1.2), we deduce that

$$\begin{aligned} \Vert \nabla F_{\sigma ,y}^{\tau _k}(x_k)-\nabla F_{\sigma ,y}(x_k)\Vert \le C(\sigma )\delta (|\tau _k-\sigma |)f(F_{\sigma }(x_k))\le C(\sigma )\delta (|\tau _k-\sigma |)f(C){\underset{k\rightarrow \infty }{\longrightarrow }}0. \end{aligned}$$

Now, if $x_k\in K(F_{\sigma })^{2\delta }$ for k large enough, as $\nabla F_{\sigma ,y}$ is proper on $K(F_{\sigma })^{2\delta }$, we deduce that up to a subsequence, we have

$$\begin{aligned} x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x_{\infty }\in K(F_{\sigma ,y})\subset K(F_{\sigma })^{\delta }, \end{aligned}$$

a contradiction. Therefore, we can assume that for all $k\in {\mathbb {N}}$) and

$$\begin{aligned} \mathrm {dist}(x_k,K(F_{\sigma }))\ge 2\delta . \end{aligned}$$

Furthermore, as $F^{\tau _k}_{\sigma ,y}=F_{\tau _k}$ and $F_{\sigma ,y}=F_{\sigma }$ on $X{\setminus } K(F_{\sigma })^{2\delta }$, we have

$$\begin{aligned} F_{\sigma }(x_k)\le F_{\tau _k}(x_k)=F^{\tau _k}_{\sigma ,y}(x_k)\le C. \end{aligned}$$

Therefore, by the Palais–Smale condition for $F_{\sigma }$, we deduce that up to subsequence, we have $x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x_{\infty }\in K(F_{\sigma })$, a contradiction. Now, to prove the second part of the claim, by (4.24), if $\tau _k{\underset{k\rightarrow \infty }{\longrightarrow }}\sigma$ and ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}\subset X$ is a sequence of critical points associated to ${\left\{ F^{\tau _k}_{\sigma ,y}\right\} }_{k\in {\mathbb {N}}}$ such that

$$\begin{aligned} F_{\sigma ,y}^{\tau _k}(x_k)\le C, \end{aligned}$$

we have by properness of $F_{\sigma ,y}$ on $K(F_{\sigma })^{2\delta }$ that (up to a subsequence) $x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x_{\infty }\in K(F_{\sigma ,y})$. Furthermore, the strong convergence of ${\left\{ x_k\right\} }_{k\in {\mathbb {N}}}$ towards $x_{\infty }$ shows that

$$\begin{aligned} \Vert \nabla ^2F^{\tau _k}_{\sigma ,y}(x_k)-\nabla ^2F_{\sigma ,y}(x_{\infty })\Vert {\underset{k\rightarrow \infty }{\longrightarrow }}0, \end{aligned}$$

(4.25)

as we see these two second order operators defined on the underlying Hilbert space $H\simeq T_{x_k}X\simeq T_{x_{\infty }}X$. Now, we recall the following continuity property of the spectrum for bounded linear operators on a Hilbert space H, which we state below.

• (P) For all $T\in {\mathscr {L}}(H)$, for all $\varepsilon >0$, there exists $\delta >0$ such that for all $S\in {\mathscr {L}}(H)$ such that $\Vert T-S\Vert <\delta$, there holds $\mathrm {Sp}(S)\subset U_{\varepsilon }(\mathrm {Sp}(T))$,

where $\mathrm {Sp}(T)\subset {\mathbb {R}}$ (resp. $\mathrm {Sp}(S)\subset {\mathbb {R}}$) is the spectrum of T (resp. S) and $U_{\varepsilon }(\mathrm {Sp}(T))$ is the $\varepsilon$-neighbourhood in ${\mathbb {R}}$ of the compact subset $\mathrm {Sp}(T)\subset {\mathbb {R}}$. Now, as $0\notin \mathrm {Sp}(\nabla ^2F_{\sigma ,y}(x_{\infty }))$, and $\mathrm {Sp}(\nabla ^2F_{\sigma ,y}(x_{\infty }))\subset {\mathbb {R}}$ is compact, there exists $\varepsilon >0$ such that

$$\begin{aligned} \mathrm {Sp}(\nabla ^2F_{\sigma ,y}(x_{\infty }))\cap (-2\varepsilon ,2\varepsilon )=\varnothing . \end{aligned}$$

(4.26)

Thanks to (4.25), for k large enough, we have by (4.26)

$$\begin{aligned} \mathrm {Sp}\left( \nabla ^2F_{\sigma ,y}^{\tau _k}(x_k)\right) \subset U_{\varepsilon }\left( \mathrm {Sp}(\nabla ^2F_{\sigma ,y}(x_{\infty }))\right) \subset {\mathbb {R}}{\setminus } (-\varepsilon ,\varepsilon ) \end{aligned}$$

so that in particular $0\notin \mathrm {Sp}\left( \nabla ^2F_{\sigma ,y}^{\tau _k}(x_k)\right)$, and $F^{\tau _k}_{\sigma ,y}$ is non-degenerate.

Finally, as $F_{\sigma ,y}$ has a finite number of critical points, and all of them are non-degenerate, this argument can be made uniform in $x_{\infty }$ and this completes the proof of the claim.

Important remark As $F^{\tau }_{\sigma ,y}-F_{\sigma ,y}=F_{\tau }-F_{\sigma }$ which is independent of y, the value $\delta (C,\sigma )$ found previously is independent of y sufficiently small.

Now, we fix some $C>\beta (1)$ and for all $\sigma \in (0,1)$, we denote $\delta (\sigma )=\delta (C,\sigma )$, and we observe that for all $j\ge J$, there holds

$$\begin{aligned} I_j=\bigcup _{\sigma \in I_j}B(\sigma ,\delta (\sigma )). \end{aligned}$$

Therefore, by compactness of $I_j=[a_{j+1},a_j]\subset {\mathbb {R}}$, there exists $N_j\in {\mathbb {N}}$ and $\sigma _1,\ldots ,\sigma _{N_j}\in I_j$ such that

$$\begin{aligned} I_j\subset \bigcup _{i=1}^{N_j}B(\sigma ,\delta (\sigma _i)), \end{aligned}$$

and up to relabelling, we can assume that $a_{j+1}\le \sigma _1<\sigma _2<\cdots <\sigma _{N_j}\le a_j$. In particular, we must have in particular $\sigma _{i+1}-\sigma _i<\delta (\sigma _i)$ for all $1\le i\le N_{j}-1$, while $\sigma _1-a_{j+1}<\delta (\sigma _1)$, and $a_{j}-\sigma _{N_j}<\delta _{N_j}$. Therefore, we define for all $y\in X$,

$$\begin{aligned} {\widetilde{F}}_{\sigma ,y}=F_{\sigma }+{\mathbf {1}}_{{\left\{ a_j\le \sigma<\sigma _1\right\} }}\varphi _{\sigma _1}\langle y,\cdot \rangle +\sum _{i=1}^{N_j-1}{\mathbf {1}}_{{\left\{ \sigma _i\le \sigma < \sigma _{i+1}\right\} }}\varphi _{\sigma _i}\langle y,\,\cdot \,\rangle +{\mathbf {1}}_{{\left\{ \sigma _{N_j}\le \sigma \le a_j\right\} }}\varphi _{\sigma _{N_j}}\langle y,\,\cdot \,\rangle \end{aligned}$$

(4.27)

where $\varphi _{\sigma _i}$ is the cut-off given by (4.23) from Proposition 2.20. Now, by notational convenience, we let $\sigma _0=a_{j+1}$ and $\sigma _{N_j+1}=a_j$.

Now, notice that $\beta (\sigma ,y)=\beta ({\widetilde{F}}_{\sigma ,y},{\mathscr {A}})$ is increasing on each interval $[\sigma _i,\sigma _{i+1}]$ for all $0\le i\le N_{j}+1$. For all $y\in X$ such that $\Vert y\Vert \le \varepsilon$, we have

$$\begin{aligned} \Vert F_{\sigma }-F_{\sigma ,y}\Vert _{C^2(X)}\le \varepsilon . \end{aligned}$$

Therefore, up to replacing $\langle y,\,\cdot \,\rangle$ by $\langle y,\,\cdot -x_i\rangle$ for some $x_i\in X$ in each component of the sum on the right-hand side of (4.27) we can assume that $F_{\sigma ,y}\ge F_{\sigma }$, so that

$$\begin{aligned} \beta (\sigma )\le \beta (\sigma ,y)\le \beta (\sigma )+\varepsilon , \end{aligned}$$

and this property of $\sigma \mapsto \beta (\sigma ,y)$ implies that

$$\begin{aligned} \int _{a_{j+1}}^{a_j}\beta '(\sigma ,y)d\sigma&=\sum _{i=0}^{N_j}\int _{\sigma _i}^{\sigma _{i+1}}\beta '(\sigma ,y)d\sigma \le \sum _{i=0}^{N_j}\left( \beta (\sigma _{i+1},y)-\beta (\sigma _i,y)\right) \\&\le \sum _{i=0}^{N_j}\left( \beta (\sigma _{i+1})-\beta (\sigma _i)+\varepsilon \right) \\&=\beta (a_j)-\beta (a_{j+1})+(N_j+1)\varepsilon . \end{aligned}$$

Taking

$$\begin{aligned} \varepsilon \le \frac{b_j}{N_j+1}, \end{aligned}$$

implies that the set

$$\begin{aligned} A_j(y)=I_j\cap {\left\{ \sigma :\beta '(\sigma ,y)\le \frac{1}{a_j\log \left( \frac{1}{a_j}\right) \log \log \left( \frac{1}{a_j}\right) }\le \frac{1}{\sigma \log \left( \frac{1}{\sigma }\right) \log \log \left( \frac{1}{\sigma }\right) }\right\} }, \end{aligned}$$

verifies by (4.22)

$$\begin{aligned} {\mathscr {L}}^1(A_j)\ge (1-2\delta _j){\mathscr {L}}^1(I_j). \end{aligned}$$

In particular, there exists for $j\ge 6\cdot 10^{702}>e^{e^{e^2}}$ an element $\sigma (y)\in I_j$ such that ${\widetilde{F}}_{\sigma ,y}$ verifies the entropy condition at $\sigma (y)$. Furthermore, as $\sigma (y)\in B(\sigma _i,\delta (\sigma _i))$ for some $i\in {\left\{ 1,\ldots ,N_j\right\} }$, we deduce that ${\widetilde{F}}_{\sigma (y),y}=F_{\sigma _i,y}$ is non-degenerate and is proper on an open neighbourhood of its critical set at level $\beta (\sigma (y),y)$, so verifies the Palais–Smale condition at this level (recall that $F_{\sigma (y),y}=F_{\sigma _{i},y}$ for some $i\in {\left\{ 0,\ldots , N_j\right\} }$, so these properties hold by Claim 1). Furthermore, as

$$\begin{aligned} \frac{d}{d\sigma }F_{\sigma ,y}=\frac{d}{d\sigma }F_{\sigma }, \end{aligned}$$

we obtain by Theorem 4.5 a critical point $x_y\in X$ of ${\widetilde{F}}_{\sigma (y),y}$ such that

$$\begin{aligned} {\widetilde{F}}_{\sigma (y),y}(x(y))&=\beta (\sigma ,y), \nonumber \\ \sigma (y)^2G(\sigma (y))&\le \frac{1}{\log \left( \frac{1}{\sigma (y)}\right) \log \log \left( \frac{1}{\sigma (y)}\right) },\quad \text {and}\;\, \mathrm {Ind}_{F_{\sigma (y),y}}(x(y))\le d. \end{aligned}$$

(4.28)

As the set of $y\in X$ such that $F_{\sigma ,y}$ is non-degenerate is dense, we can choose a sequence ${\left\{ y_k\right\} }_{k\in {\mathbb {N}}}\subset X$ such that $y_k{\underset{k\rightarrow \infty }{\longrightarrow }}0$, such that $F_{\sigma _i,y_k}$ is non-degenerate for all $1\le i\le N_j$ for all $k\in {\mathbb {N}}$, and $\sigma (y_k)=\sigma _k^j\in I_j$ such that ${\widetilde{F}}_{\sigma _k,y_k}$ admits a critical point $x(y_k)=x_k\in X$ verifying (4.28). As $I_j$ is compact, we can assume up to a subsequence that $\sigma _k^j{\underset{k\rightarrow \infty }{\longrightarrow }}\sigma _{\infty }^j\in I_j$, and as

$$\begin{aligned} \Vert {\widetilde{F}}_{\sigma _k,y_k}-F_{\sigma _k}\Vert _{C^2(X)}{\underset{k\rightarrow \infty }{\longrightarrow }}0, \end{aligned}$$

(4.29)

we deduce that up to a subsequence, by the Energy bound (2) and (4.29), we have (notice that $\nabla F_{\sigma _k,y_k}(x_k)=0$)

$$\begin{aligned} \Vert \nabla F_{\sigma }(x_k)\Vert&\le \Vert \nabla F_{\sigma }(x_k)-\nabla F_{\sigma _k}(x_k)\Vert +\Vert \nabla F_{\sigma _k}(x_k)\Vert \\&=\Vert \nabla F_{\sigma }(x_k)-\nabla F_{\sigma _k}(x_k)\Vert +\Vert \nabla F_{\sigma _k}(x_k)-\nabla F_{\sigma _k,y_k}(x_k)\Vert {\underset{k\rightarrow \infty }{\longrightarrow }}0. \end{aligned}$$

Therefore, up to an additional subsequence and by the Palais–Smale condition, we have the strong convergence

$$\begin{aligned} x_k{\underset{k\rightarrow \infty }{\longrightarrow }}x_{\infty }^j\in K(F_{\sigma _{\infty }^j}). \end{aligned}$$

Finally, by the strong convergence of the second derivative, we have

$$\begin{aligned} \mathrm {Ind}_{F_{\sigma _{\infty }^j}}(x_{\infty }^j)\le \liminf _{k\rightarrow \infty }\mathrm {Ind}_{F_{\sigma _{k},y_k}}(x_k) \end{aligned}$$

(4.30)

and (notice that by non-degeneracy of $x_k$ for $F_{\sigma _k,y_k}$ that $\mathrm {Null}_{F_{\sigma _k,y_k}}(x_k)=0$)

$$\begin{aligned} \mathrm {Ind}_{F_{\sigma _{\infty }^j}}(x^j_{\infty })+\mathrm {Null}_{F_{\sigma _{\infty }^j}}(x^j_{\infty })\ge \limsup _{k\rightarrow \infty }\left( \mathrm {Ind}_{F_{\sigma _{k},y_k}}(x_k)+\mathrm {Null}_{F_{\sigma _k,y_k}}(x_k)\right) \end{aligned}$$

(4.31)

so $x^j_{\infty }$ verifies (4.28) for $y=0$ and $\sigma (y)=\sigma _{\infty }^j$. Furthermore, if ${\mathscr {A}}$ is replaced by a dual family, then the one-sided estimate from below of the index is given by (4.31) while two sided estimates are given for co-dual, homological or cohomological families by (4.30) and (4.31).

This concludes the proof of the theorem, as the sequences ${\left\{ \sigma _{\infty }^j\right\} }_{j\in {\mathbb {N}}}\subset (0,\infty )$ and ${\left\{ x_{\infty }^j\right\} }_{j\in {\mathbb {N}}}\subset X$ satisfy the conditions of the theorem. $\square$