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Prescribing Morse scalar curvatures: incompatibility of non existence

Given a closed manifold of positive Yamabe invariant and for instance positive Morse functions upon it, the conformally prescribed scalar curvature problem raises the question, whether or not such functions can by conformally changing the metric be realised as the scalar curvature of this manifold. As we shall quantify depending on the shape and structure of such functions, every lack of a solution for some candidate function leads to existence of energetically uniformly bounded solutions for entire classes of related candidate functions.


Introduction
Give a closed and smooth Riemannian manifold M of dimension n ≥ 2, the prescribed scalar curvature problem is concerned with the question, which functions K on M can be prescribed as the scalar curvature R = R g on M for some metric g on M , i.e. solvability of R g = K.
If we allow to choose any metric, we wish, this problem has been solved in [15], [16]. If we restrict the variation to a conformal class of metrics, i.e. we consider M = (M, g 0 ) and are allowed to only choose among metrics g conformal to g 0 , i.e. g ∈ [g 0 ] the conformal class of g 0 , we refer to the conformally prescribed scalar curvature problem. Geometrically of course the restriction to a conformal class is natural, since the Cotton tensor for n = 3 and the Weyl tensor for n ≥ 4 are invariant under a conformal change of the metric. So we consider the conformal problem exclusively.
Analogously to the Yamabe case of constant K, the conformally prescribed scalar curvature problem decays into three categories according to the sign of the conformal Yamabe invariant, i.e. , where µ g denotes the measure on M induced by g. We focus on K, Y > 0.

The Positive Case
The equation to solve is of variational nature, i.e. solutions to the conformally prescribed scalar curvature problem R gu = K > 0 correspond to critical points of an energy J K , which in this case is positively lower bounded against the positive Yamabe invariant, cf. Section 4. Secondly the equation is critical with respect to Sobolev embeddings and therefore sublevel sets of J K are generally non compact. To compensate this, one may consider symmetric or even radial situations to at least recover partial compactness, cf. [10], [14], or pass to ordinary differential equations methods directly, cf. [13].

Lack of Compactness
Generally however, following some pseudo gradient flow Φ, we may lack convergence towards a critical point. This lack of compactness then leads to a blow up, which is described by bubbling. In the simplest case and for n ≥ 3 such a flow line u develops a single bubble cf. [20], [21], and with positive constants c 1 , c 2 , c 3 > 0 the energy expands as where H ≥ 0 is a geometric quantity, referred to as mass, cf. [26], and Heuristically we then may decrease the energy J K by decreasing the remainder term v and, as long as |∇K(a)| = 0, we have a clear direction to decrease energy, namely moving a along ∇K(a). If we come close to a region, where ∇K = 0, the following, non exhaustive situations may occur.
These three scenarios display the relation of the shape of K at critical points, represented by ∆K = 0, to the geometry there, represented by H in dependence of the dimension, namely subordinate, balanced and dominated. Moreover the dimension does not only affect the blow up behaviour for single bubbles, but also the question of multi bubbling, cf. [18], [24], [27], the question, whether or not, non pure bubbling may occur, cf. [4]. Note, that the exclusion of non simple blow ups, i.e. multi bubbling at some point, is a given in case of closed manifolds, cf. [18], [21], but happens in other geometric situations, cf. [2].

The Morse case
Let us briefly discuss a few relevant results in the case, that K > 0 is a generic Morse function satisfying the classical, mild non degeneracy condition n=2 This is the case of M ≃ S 2 . In [11] we find an application of the Mountain Pass theorem under the assumption, that K does have at least two maxima, while Connecting the latter maxima, which correspond to critical points at infinity of index 0, by a path leads us to a criticality of index 1, which cannot stem from a critical point at infinity due to (2). Secondly in [12] and denoting by p = ♯{K = max K} and q = ♯{∇K = 0 : m(K, ·) = 1} the number of maxima and saddle points respectively, we find existence for This corresponds to the results in [6] for S 3 and may be expressed as with χ(X) the Euler characteristic of the contractible variational space X, the set of critical points at infinity and the index of a critical point at infinity c ∈ C ∞ (J K ). Hence (3) and (4) are equivalent and clearly the right hand side of (4) should yield the Euler characteristic of X in absence of any critical point.
n=3 In case M ≃ S 3 the analogous results to M ≃ S 2 , as described above, were proven in [8] for the Mountain Pass argument and in [6] for the index counting formula (4), cf. Section 5. If M ≃ S 3 , the problem becomes compact on low sublevels and existence of a minimizer is immediate, cf. [14].
n=4 We refer to [8] for a restrictive application of the Mountain Pass theorem and to [7] for a delicate analysis of the the lack of compactness and a resulting index counting formula.
n ≥ 5 Mountain Pass results are available under restrictive assumptions, cf. [8], [19], while an index counting formula is not available, cf. Lemma 5.2. In case of Einstein manifolds and under pinching, existence is assured, provided K admits more than one critical point of negative Laplacian. There is also another invariant, besides the Euler characteristic, which assures existence, cf. [4], [28]. However, as far as we are aware of, its computation requires assumptions, which are difficult to verify.

Ambiguity
To illustrate our main result, Theorem 3.2, consider the smooth heart S ≃ S 2 on S 3 , cf. Figure 1 and Section 5.2 for further discussions.
x 0 x 1 Figure 1: The smooth heart as a height function Precisely for a function K > 0 on S 3 we imagine for K −1 = 1 K exactly six critical points, which are (i) two local maxima x 1 3 , x 2 3 ∈ S 3 , one inside and one outside S of index They are not depicted in Figure 1.
The height in Figure 1 is to visualise their relative values on S ⊂ S 3 , i.e.
Due to the index counting formula from [6] and by Theorem 5.3 we can solve the corresponding conformally prescribed scalar curvature problem, unless for the minimum x 0 and highest value saddle point x 2 2 of K −1 and These are two scenarios, essentially distinguished by the sign of the Laplacian at critical points, for which we cannot conclude existence of solutions. But Theorem 3.2 or rather its 3-dimensional version tells us, that at most one of these cases is not solvable. And unfortunately we do not know, which one, since the proof of Theorem 3.2 is non constructive.
Generally Theorem 3.2 shows, that there is in a certain local sense always at most one combination of critical points with negative Laplacian such, that we cannot assure existence of solutions.

Remarks
We proceed as follows. After lengthy constructions in Section 2, required for the sake of clear statements, we present in Section 3 a deformation scheme, which applied to variational functionals of the conformally prescribed scalar curvature problem leads to Theorem 3.2. In Section 4 we quickly review the variational setting of the critical equation and the relation to its subcritical approximation. Then, in order to clarify the theoretical interest and limitations of Theorem 3.2, we describe in Section 5 the classical total degree or, as it is called, the index counting formula argument for existence and discuss its failure. Here we also study the smooth heart on S 3 and demonstrate Theorem 5.3. In Section 6 we then show, how to exactly apply the aforementioned deformation scheme leading to Theorem 3.2. Finally in the Appendix we include for the sake of completeness some topological lemmata, on which the deformation scheme is based, as well as a standard remark on Morse functions, which shows generic non triviality of Theorem 3.2.
Before passing to the actual content, we point out, that the subsequent arguments, which we provide for n ≥ 5, do have at least for n = 2, 3 less stringent adaptations and are actually easier to obtain, as we shall comment on here and there. For the Gaussian case n = 2 however we refer to [23] with stronger results.

Partition and Classes
Let us start with the following construction. Consider sequences . . < ∞ and some fixed indices we have upon reordering, cf. (6).
We say, that elements of M for all K ∈ M.
Note, that in case n ≥ 5 and for a function K ∈ M +,S,k,k i1,...,im (M ) the set determines the pure critical points at infinity of J K , cf. Section 4. In fact for every subset there exists exactly one critical point at infinity of type with energy, cf. (8), Likewise for n = 2, 3, but replacing subsets with elements due to single bubbling, and also the conditions (8) and (10) can be relaxed from subsets to elements.

Remark 2.2.
Here some comments are in order to clarify (i)-(v) above.
(i) is the usual non degeneracy assumption, which for Morse functions allows us to identify the critical points at infinity easily, cf. [21].
(ii) Given a Morse function K on M , consider the space L K of Morse functions on M , whose critical points correspond one to one to those of K with coinciding Morse indices. Then a Morse structure on M is just any space L K induced by some Morse function K.
(iii) tells us, that irrespective of a particular choice of K the critical points, which potentially induce a single bubbling critical point at infinity by having a negative Laplacian, are distinguished by being contained in different energy strips, cf. (7) and (12).
(iv) is to say, that irrespective of a particular choice of K all potential multi bubbling critical points at infinity are distinguished by being contained in different energy strips.
(v) additionally assumes, that for a multi bubbling critical point at infinity the energy strip, in which it is contained by (iv), uniquely determines the critical points of K involved, cf. (10) and (12).
(vi) Generally a sufficiently small C 0 -neighbourhood of a spreading function K contains an abundance of functions satisfying (i)-(v) with varying signs of the Laplacian at non extremal critical points, cf. Appendix 7.2. Hence every spreading function K creates locally some spread M with K ∈ M.
We then associate to each spreading function its critical points with negative Laplacian and obtain from (7) a map N : M and denote byM for a spread M ⊆ M +,S,k,k i1,...,im (M ) the induced quotient. Remark 2.4. Definition 2.3 is just to say, that we identify those functions, whose critical points with negative Laplacian correspond one to one as unique elements of the same energy strip.
The impact of (10) is, thatM becomes an ordered set as. In fact consider for K ∈ M, M a spread, the maximal energy value over all pure critical points at infinity of J K , i.e. those of type (11). Since n ≥ 5, the maximal value µ(K) arises as the energetic value of the unique pure critical point at infinity constituted of all the critical points of K with negative Laplacian, i.e.
We then obtain the desired ordering, as follows, from the maximal energy µ, while we postpone the proof to the Appendix.
Lemma 2.5. Let K 1 , K 2 ∈ M, M a spread. Then In particular the partition ∼ on M leading toM defines an injective mapping As a consequence we may order the equivalence classesK ∈M viã which is just to say, that we induce an ordering onM according to energy strip, in which we find the maximal pure critical value at infinity of representatives. Hence and conversely we have a bijection onto the set of equivalence classesM, which we index as accordingly. With this notions at hand, we make Hence by definition and recalling N ⊂ N ≥1 In other wordsK σ is, if existent, the energetically highest class of functions in a spread M, for which we cannot solve for at least one representative K σ ∈K σ .
Remark 2.7. In case n = 2, 3 we may likewise partition along Due to single bubbling and by (7) or equivalently (v) in Definition 2.1 this means, that we simply identify those function K, whose minimal values min x∈C−(K) K over their critical points with negative Laplacian lie in the same energy strip, cf. Definition 2.1,(iii). Thus we identify much more functions K with respect to the case n ≥ 5, since there functions are identified, if all their critical points with negative Laplacian lie in the same energy strips. However, as we will see later on in Proposition 6.1, the deformation scheme is applied to the highest energy critical point at infinity, which in case n = 2, 3 corresponds to the unique critical point of K with negative Laplacian realising min x∈C−(K) K.

A Deformation Scheme
We present a deformation scheme, leading by means of a comparison argument to the existence of critical points of a functional. The scheme is non constructive, as it follows from arguing by contradiction. The contradiction would be, that in absence of a critical point, whose existence we wish to prove, it was possible to circumvent by energetic deformation an energetically isolated, non degenerate critical point of a different functional, which topologically is impossible. We visualise the setting in Figure 2.  (ii) J has exactly one non degenerate critical point for a neighbourhood U c of the unstable manifold W u (c).
(iii) All critical points of J in D are non degenerate and there holds Then necessarily Proof. Arguing by contradiction we may retract by weak deformation cf. Corollary 7.2. On the other hand we have weak deformation retracts Then from (15) and (16) we deduce the homotopy equivalence B ≃ B, while attached, which is an obvious contradiction and (14) follows.
Applying Proposition 3.1 for σ < σ and suitable M to the functionals for some 0 < κ < 1 < κ < ∞, there exists at most one classK σ ∈M such, that for at least one representative K ∈K σ is possible.
Proof. Recalling Definition 2.6 and the remarks thereafter, first note, that in cases σ = 0 or σ = min N there is nothing to show. Hence we may assume, that there exists σ < σ with σ, σ ∈ N . We then have to show But this follows from Proposition 6.1, provided the conditions And we simply choose κ σ−1 = κ and κ σ = κ.  [19] is guaranteed, provided more than one critical point has negative Laplacian. Then, 1.) if S provides at least two local maxima, then existence of a solution to ∂J K = 0 is always guaranteed, which is equivalent to σ = 0.
2.) if S provides one maximum, then either σ = 0 or, if σ ≥ 1, then necessarilyK σ is the class of functions, whose only critical point with negative Laplacian is the global maximum.
Clearly Theorem 3.2, while non constructive, is nonetheless a mean to turn an obstruction to existence for one specific function into existence results for entire classes of comparable functions. And likewise Theorem 3.2 is a mean to turn the lack of low energy solutions for one specific function into an a priori energy bound for solutions of entire classes, whose existence either follows from Theorem 3.2 or by contradiction.
In particular Theorem 3.2 (i) renders the quest of establishing non existence results a very fruitful endeavour, cf. [9], with the caveat, that to run the deformation scheme always requires a description of its critical points at infinity.
(ii) tells us, that depending on the complexity of a Morse structure most positive scalar curvature candidates realising this structure are conformally prescribable in a controllable way.
(iii) strongly hints at deeper structures on the space of solutions to the conformally prescribed scalar curvature problem.

The Variational Formulation
Let us review the variational setting, while we refer to [17], [18], [21] for greater detail. Consider a closed Riemannian manifold M = (M n , g 0 ) with n ≥ 5, of positive Yamabe invariant, cf. (1), induced volume measure µ g0 and scalar curvature R g0 > 0. In this setting the conformal Laplacian is a positive and self-adjoint operator and considering a conformal metric So prescribing conformally the scalar curvature as a given function K is equivalent to solving Due to R g0 > 0 evidently defines an equivalent norm on W 1,2 . We then wish to study on the scaling invariant functional Note, that J K is scaling invariant, which allows us the consider as a natural variational space. Moreover the conformal scalar curvature We then find there is, cf. [21], a one to one correspondence to the pure critical points at infinity of J K , i.e. those of type where δ a denotes a Dirac measure in L 2n n−2 at a ∈ M .
Finally, if {∂J K = 0} = ∅, these are the only critical points at infinity 1 .

Subcritical Transition
Let us translate these notions into the realm of subcritical approximations. As exposed in [18] there is under (18) an analogous one to one correspondence from P(C − (K)) to the zero weak, finite energy limiting critical points of the subcritical approximation functional These critical points are of isolated simple bubbling type (τ, {x 1 , . . . , x q }) −→ u τ,x1,...,xq = α 1 ϕ a1,λ1 + . . . + α q ϕ aq,λq + v, where ϕ a,λ denotes a suitable bubble, and have critical limiting energy We will always assume (18) and in fact rely in our arguments on this close relationship of pure critical points at infinity and zero weak, finite energy limiting subcritical solutions. In particular the following non existence statement is analogous to the "only" statement on critical points at infinity in case {∂J K = 0}.
In case q = 0, i.e. u τm = u ∞ + v m , we then have In case q > 0 and u ∞ > 0 we have by positivity of L g0 . Also, as follows from testing 0 = ∂J τm (u τm ) with u ∞ , while due to (21). Then from (22), (23) and (24) we deduce and, as there are only finitely many, they induce a change of topology of the sublevel sets J c±ε K,τ = {J K,τ ≤ c ± ε} by attaching cells D i with dim(D i ) = m(J K,τ , u τ,xi 1 ,...,xi q ).
And likewise we have a homological equivalence for the pure critical points at infinity, cf. [21].

Obstructions to Non Existence
Let us discuss in case of a positive Yamabe invariant some standard arguments to prove existence of solutions to ∂J K = 0, i.e. the conformally prescribed scalar curvature problem. In fact their failure does shed some light on, why Theorem 3.2 is of theoretical interest.

The Euler Characteristic
Assuming non existence, i.e. {∂J K = 0} = ∅, we have contractibility of cf. (13). Indeed we may first for some ε > 0 retract by strong deformation and then perform a star shaped homotopy which is well defined, provided L ≫ µ(K) is sufficiently large. We conclude Then J L K is contractible for all L ≫ µ K sufficiently large, provided In particular χ(J L K ) = 1 for the Euler characteristic. On the other hand we may compute the latter Euler characteristic by Morse theory and obtain under the assumptions of Lemma 5.1 Generally this identity may be violated, for instance in low dimensions or for higher flatness of K near its critical points and in this case, arguing by contradiction, necessarily at least one solution to ∂J K = 0 has to exist, cf. [6], [7], [12], [24], [25]. The underlying reason is, that in these cases C ∞ (J K ) corresponds to a proper subset Q P(C − (K)), i.e. not every combination gives rise to a critical point at infinity. In our setting, i.e. under (18), however the index counting formula (26) always 2 holds true.  We then choose y ∈ {K = max K} ⊆ C − (K) and decompose We then have a bijection  m(K, y))). In particular, since y is a maximum of K, we have m(K, y) = n and obtain ind(J K , u ∞,x1,...,xq,y ) = ind(J K , u ∞,x1,...,xq ) + 1.
We thus find cancellation of C 2 with C 3 in the sense, that Hence this standard argument for existence based on the computation of the Euler characteristic by showing contractibility on the one hand and counting Morse indices on the other is not always feasible, which is to say, that the computation of the Euler characteristic does not provide an obstruction to non existence.
However, even if (26) holds or in other words, if the total degree is zero, one may still obtain existence of solutions by contradiction arguments, if (i) some finer homological identities would is violated, cf. [3].
(ii) a min-max scheme leads to a critical point at infinity of certain indices, which is not there, cf.
(iii) the comparability of sublevels of different functionals are incompatible with non existence assumptions, as we argue here and in [19].

Connecting Orbits
Let us consider the smooth heart S ≃ S 2 as depicted in Figure 1 on S 3 , which is perhaps the most simple, yet ambiguous example, cf. the Introduction. Every argument clarifying this situation will -in our opinion -be a cornerstone in the future development, not only within, but also beyond the realm of scalar curvature problems.
Let us describe upon a choice of orientation the Morse homology over Z 2 .
To see (27) note, that x 1 2 , x 2 2 have each exactly one positive direction orthogonal to S pointing inand outside of S. Following these directions, we must from both x 1 2 and x 2 2 arrive with a unique, single orbit at each maxima x 1 3 , x 2 3 , cf. (5). As a consequence 3) H 3 (S) = ker∂3 /Im∂4 = ker∂ 3 = Z 2 , since clearly Im∂ 4 = 0 and by (28) Passing to the prescribed scalar curvature functional J K , suppose, that we cannot solve ∂J K = 0, while K > 0 Morse satisfies Then by [6] the critical points at infinity of J K correspond exactly to single Diracs at elements of Since ∆K(x 0 ) < 0 < ∆K(x 1 3 ), ∆K(x 2 3 ) due to (29), (26) leaves us with as possibilities for non existence and we rule out 3.) in the following theorem.
Theorem 5.3. For the smooth heart S on S 3 suppose, that Passing to the subcritical problem, first note, that for u τ,x 2 2 ∈ {∂J K,τ = 0} and τ ≫ 1, cf. (19), there holds for a suitable choice of λ ≃ τ − 1 2 ≫ 1. This clearly holds in case of strict inequality in (i), but also follows easily in case of equality by expansion from Proposition 5.1 in [17] using, that there (i) ε i,j = 0, as q = 1 With this in mind, we first prove the existence of a solution In fact and arguing by contradiction suppose By virtue of (30) we then may inject by bubbling for some σ < J K,τ (u τ,x 2 2 ) i : for suitable, continuous maps x −→ α x , a x , λ x , v x and under this map are homeomorphic. In fact (33) is immediate from the Morse structure of K around x 1 and of J K,τ around u τ,x1 , cf. [18]. On the other hand as a weak deformation retract, if we deform {J ≤ σ} along 1) the gradient flow locally patched with a strong deformation retract onto 2) W u (u τ,x1 ) close to u τ,x1 leaving W u (u τ,x1 ) invariant and onto 3) u τ,x0 induced by flowing close to, but not at u τ,x0 along which leaves W u (u τ,x1 ) invariant as well.
But then from (32) and (33) we obtain a continuous map homotopic to the identity on S 1 ≃ W u (u τ,x1 ) in contradiction to Brouwer's fixed point theorem. Hence (31) and letting τ −→ 0 the assertion follows.
Remark 5.4. We remark, that (i) K(x 1 2 ) = K(x 2 2 ) and therefore coinciding potentially critical values for J K , are permitted in Theorem 5.3, but not in Theorem 3.2.
(ii) there is another way of seeing Theorem 5.3. Arguing by contradiction, the functional J K,τ is Morse with exactly two Morse homological cycles c 0 , c 1 ∈ {J K,τ ≤ σ} on the latter set, which are induced by u τ,x0 and u τ,x1 . In particular for the first Morse homology group. But Morse and singular homology coincide, since the critical points of J K,τ have finite indices, cf. [1]. And in singular homology the only possible cycle is up to homotopy the unstable manifold of u τ,x1 , which is however the boundary of the 2-cell Hence H 1 ({J K,τ ≤ σ}) = 0 and this is a contradiction.
Theorem 5.3 leaves us with the possibilities 1.) and 2.) and both are variationally stable, as follows.
1.) In this case u ∞,x0 is the only critical point at infinity of J K , which acts as a minimum. Correspondingly u τ,x0 is a minimum for the subcritical approximation J K,τ and the only energy bounded, zero weak limit critical point.
2.) Also in this case we see variational stability at infinity for the functional J K as well as for its subcritical approximation J K,τ .
(i) Consider some σ < J K (α x 2 2 δ x 2 2 ) and for 0 < ε ≪ 1 Note, that the latter set is a neighbourhood of the critical points at infinity u ∞,x0 , u ∞,x1 and u ∞,x 1 2 . Given thus u 0 as an initial data consider a flow line Then, as long as u remains in some neighbourhood we have with positive constants c 1 , c 2 , c 3 > 0 . In fact (α) follows from (100) in [6], since 0 < c < α < C due to u = 1, and by scaling invariance while we have the estimates and, as follows by direct calculation, Secondly (a),(λ) follow from Lemma 6 in [6] using again (34) and (35) above to estimate |∂J(u)|. Finally to verify (v) we calculate using (35) above and from [6] Lemma A2 and (Vo).

Hence (α),(a),(λ) and (v) are established and from (v) we deduce
as long as u ∈ V (1, ǫ). And during that time by (a) and (λ) since ∆K(a) < 0 close to x 0 , x 1 , x 1 2 , which for energy reasons are the only reachable critical points of K. Hence λ cannot substantially decrease and we may assume as long as u ∈ V (1, ǫ). But then from (36) and (37) we conclude, that u can actually never leave V (1, ǫ). Moreover from (37) and (a) we then find, that a converges to x 0 , x 1 or x 1 2 , where by (λ) evidently λ −→ ∞ and due to (36) also v −→ 0.
We summarise this by saying, that full neighbourhoods of all the pure critical points at infinity remain concentrated along the negative gradient flow. In particular any flow line leading from one of them to the other does so in a concentrated way.
(ii) An analogous argument based on the evolution of α, a, λ and v under ∂ t u = −∇J K,τ (u) will show, that also some neighbourhoods and then also the unstable manifolds of u τ,x0 , u τ,x1 and u τ,x 1 2 remain in some V (1, ǫ), i.e. remain of type u = αδ a,λ + v and of course one may choose ǫ −→ 0 as τ −→ 0.
This tells us, that the Morse complex, which these critical points induce, is completely and classically describable and this without any homological inconsistency.
We leave this claim without a proof, since the concrete computations would go beyond the scope of this paper. However note, that transversality of the intersection of the stable and unstable manifolds is generic, since u τ,x0 is a minimum and m(J K,τ u τ,x 1 2 ) − m(J K,τ u τ,x1 ) = 1 for the relative Morse index, cf. [1]. Now, while neither for K 1 satisfying 1.) nor for K 2 satisfying 2.) we see any obvious obstruction to non existence, we may ask, whether assumed non existence for both K 1 and K 2 is compatible, i.e. whether leads to a contradiction, hence establishing And this is exactly, what Theorem 3.2 addresses.

The Comparison
Recall, that by construction, cf. (13) and Lemma 2.5 with the remarks following, the maximal pure critical value and arise from corresponding critical points at infinity Moreover, as follows as Proposition 4.1, if then the only critical points at infinity within {J ≤ L} are those pure type and hence the maximal critical and the maximal pure critical value on the latter set coincide.
Recalling Definition 2.6 and the following comments, we then suppose and wish for someσ < σ to lead for a suitable choice of L ≥ 0 to a contradiction.
Since by construction we are then in the situation, that all critical values of J Kσ , i.e.
and all critical values of J Kσ apart from µ(K σ ), i.e.
the unique critical point at infinity of J Kσ with maximal energy and let us accordingly denote by cf. Section 4.1, the critical point of J Kσ ,τ associated to u ∞,σ .
Now for 0 < τ ≪ 1 fixed let us translate these relations to J = J Kσ ,τ and I = J Kσ,τ , cf. Proposition 3.1. Clearly J and I satisfy by virtue of (40) Moreover and recalling (38) and 1.) above let us also choose a suitable and denote by Also by (40) there holds Let us now show for We first observe, that obviously J and I satisfy the Palais-Smale condition, while the critical points of J in D are all non degenerate, cf. Section 4.1, and is the unique critical point of J therein, since without loss of generality We are thus left with verifying Indeed (i) D ⊇ C follows by (41).
(iii) B ⊇ B is trivial.

Appendix
Proof of Lemma 2.5. To show the Lemma, we suppose (i) on one hand, that Denoting by Letting N = {j l : 1 ≤ l ≤ l 1 } the desired conclusion follows from (10).
(ii) on the other hand, that for some i ∈ N which translates by (8) for the subsets Then (10) implies N 1 = N 2 and the conclusion follows.
This proves the desired equivalence statement.

Homotopy Equivalences
We derive some elementary results on homotopy equivalences, denoted by X ≃ Y for two topological sets X and Y , i.e. the existence of continuous mappings where we denote by φ ≃ ϕ the existence of a homotopy Proof. Let us first recallg and note, that proving A ≃ D is sufficient. To that end consider We then have Hence F with inverse G provides a homotopy equivalence A ≃ D.
An easy, yet useful consequence is satisfyingĝ we then haveĝ i.e. A ≃ C underf , whilê
ClearlyK(x 0 ) = K(x 0 ), and there holds on B 2δ (x 0 ) In particular choosing for some fixed ǫ > 0 ∀ 1 ≤ j ≤ n : |c j − b j | ≤ ǫ|b j | we find, that K −K C 0 = O(εδ 2 ) and (iv)K is a Morse function as is K (v) close to any x ∈ {∇K = 0} there holds W K u (x) = WK u (x) and W K s (x) = WK s (x) for the stable and unstable manifolds W u , W s of K,K of x.
Note, that the value of ∆K at a non extremal critical point may be made positive or negative, cf. (47). And this shows the local non triviality of sets as in Theorem 3.2.