Prescribing scalar curvatures: non compactness versus critical points at infinity

We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.


Introduction
Within the setting of conformally prescribing the scalar curvature on a Riemannian manifold and in the context of the calculus of variations, i.e. by considering an associated energy functional, we shall illustrate in a very particular case the difference of non compact flow lines of a given gradient flow to critical points at infinity, as we have discussed in [16], namely showing, that the volume preserving L 2 -gradient flow (1.1), which is a natural analogon to the Yamabe flow and was studied in [15], exhibits one specific, single bubbling non compactness for exactly one energetic value of the variationally associated prescribed scalar curvature functional, while a suitable modification of this flow eliminates any non compactness.
And, as we shall see, the same holds true for the strong gradient type flow (1.3) modified to preserve the conformal volume just like (1.1). Hence as a take away those non compact flow lines do not induce critical points at infinity, cf. [16], i.e. these flows lead to variationally unmotivated singularities and are hence as geometric flows evidently not the best choice in the context of the calculus of variations, i.e. for energetic deformations.
However such gradient type flows, whether weak or strong, i.e. with respect to a L 2 -or W 1,2 -gradient, are of interest in their own right apart from their usefulness in proving mere existence results to the underlying elliptic problem of prescribing the scalar curvature on a Riemannian manifold conformally, in particular due to the naturality of L 2 -gradient flows for a geometric problem.
We wish to mention some works relevant to the flow analysis.
(i) The most simple case evidently is, when the function K to be prescribed is constant, e.g. K = 1, and the underlying manifold is the standard sphere S n , in which case flow convergence is known, cf. [2], [19], with exponential speed, cf. [7].
(ii) Later on and based on the positive mass theorem also on non spherical manifolds flow convergence in the Yamabe case K = 1 was established, cf. [19], [18], [8], with a subsequent analysis on upper and lower bounds of the speed of convergence, cf. [9].
(iii) Returning to the spherical case M = S n , but considering a non constant function K to be conformally prescribed as the scalar curvature, flows and their lack of compactness were first analysed and characterised in [2], [3] and [5] in case n = 3. For higher dimensional cases we refer to [6] for n = 4 and to [16] for n ≥ 5, see also [12], [13] and [14].
(iv) Finally the case of a general Riemannian manifold M with non constant K to be prescribed, to which the present work belongs, has been less studied with respect to an analysis of gradient flows. We point in case of a positive Yamabe invariant of M to [16] for a classification of non compactness in dimensions n ≥ 5 and to [15] for some compactness results in dimensions n = 3, 4, 5. In case of a negative Yamabe invariant flow convergence was proven in [1] recently.
In order to introduce the relevant notions, consider a smooth, closed Riemannian manifold is a positive, selfadjoint operator with Green's function G g0 . We may assume R g0 > 0 and Kdµ g0 = 1 for the background metric g 0 . For a conformal metric g = g u = u 4 n−2 g 0 there holds dµ = dµ gu = u 2n n−2 dµ g0 for the volume element and R = R gu = u − n+2 n−2 (−c n ∆ g0 u + R g0 u) = u − n+2 n−2 L g0 u for the scalar curvature. We may define and use · as an equivalent norm on W 1,2 (M ). Let 0 < K ∈ C ∞ (M ) and r = r u = Rdµ, k = k u = Kdµ,K =K u = K k .
In [15] we have studied the L 2 -pseudo gradient flow which evidently coincides with the Yamabe flow in case K = 1. Obviously ∂ t k = 0, i.e. the unit volume k ≡ 1 is preserved. Let us consider the scaling invariant energy J(u) = c n |∇u| 2 g0 + R g0 u 2 dµ g0 ( Ku Moreover J is C 2,α loc and uniformly Hölder continuous on each In particular the problem of conformally prescribing the scalar curvature is variational and Then by a slight abuse of notation we define as a natural majorant of |∂J(u)| and along a flow line we have From Theorem 1 in [15] we know at least in cases n = 3, 4, 5, that every flow line for (1.1) exists positively for all times. Consequently we have a priori as J by positivity of the Yamabe invariant is lower bounded. Similarly we may consider the gradient flow This describes a strong gradient flow, since by definition and we write ∇J(u) = L − g0 ∂J(u). For the sake of easy comparability to (1.1) consider as a strong pseudo gradient flow. Then ∂ t k = 0 and, since by scaling invariance we have ∂J(u)u = 0, In particular and by positivity of the Yamabe invariant we have along each flow line so u > 0 is preserved. Indeed due to k = 1 and (1.4) we find from Proposition 1.1, that |∂J(u)| is a priori bounded along flow lines. Therefore each flow line exists positively for all times and We thus see, that (1.3) defines a pseudo gradient flow on X as well. Note, that (1.3) falls into the class of ordinary differential equations, hence long time existence is a non issue in contrast to the L 2 -type flow (1.1). The difference, when considering (1.1) in contrast to (1.3) apart from the distinguishing quadratic a priori integrability of |δJ| versus |∂J| lies in the ease of adaptability. In fact considering a bounded and for instance smooth vectorfield W on X satisfying ∇J, W ≥ 0 we may modify (1.3) to as we shall do in Section 3.3. We then still decrease energy, find quadratic a priori integrability of |∂J|, preserve ∂ t k = 0 and u > 0 and finally also (1.6) falls into the class of ordinary differential equations, hence also (1.6) defines a flow on X. In contrast the long time existence of (1.1) relies on higher order integrability properties of R − rK, cf. [8], [15], which may be destroyed by even slight adaptations.
In any case, i.e. (1.1),(1.3) or (1.6), the volume k = 1 is preserved and the lower bounded energy J decreased, whence along a flow line u i.e. we have norm control along each flow line. Moreover under (1.1) there holds cf. Proposition 2.11 in [15]. Likewise there holds under (1.6) for a least a sequence t k −→ ∞ as k −→ ∞ in time and thus for any using a priori uniform boundedness of |∂J(u)| and |∂ 2 J(u)|, cf. Proposition 1.1, along flow lines.
Based on a fine description of a possible lack of compactness of flow lines, we had extracted suitable assumptions to guarantee compactness of the flow on X induced by (1.1), cf. Theorem 2 from [15]. For instance for n = 5 under in a conformal normal coordinate system around x 0 ≃ 0 we have We refer to [11] and [10] for the notion of conformal normal coordintates. Also note, that we only slightly violate Cond 5 , since indeed close to x 0 we have in particular Cond 5 from [15] guaranteeing flow convergence is pretty sharp. As a consequence the only possible non compactness, i.e. non compact flow lines for (1.1) or (1.3), correspond to a bubbling close to x 0 with critical energy . This unique bubbling then occurs both for (1.1) and (1.3) and we will compare these flows in detail. However by a slight modification of the latter flow in the spirit of (1.6) this non compactness will be completely removed. for the prescribed scalar curvature functional (1.2) exhibit exclusively non compact flow lines of single bubble type at the unique maximum of K, while there exists a compact pseudo gradient for the latter functional, i.e. a pseudo gradient, all of whose flow lines are compact and hence converging.
Proof. We have seen above, that (1.1) and (1.3) induce a flow Φ on X, whose flow lines up to a time sequence are Palais-Smale. Then up to a subsequence in time Hence convergence in case p = 0. By Section 3.1 only p = 1 is possible in case p > 0 and then Lemma 3.4 then shows, that indeed λ −→ ∞ for suitable initial data. Hence we have proven the exclusive existence of non compact flow lines as a single bubbling at x 0 . Finally for the modified flow on X induced by (3.17), which is a pseudo gradient flow by virtue of Lemma 3.5, the only possibility for a non compact flow line is as before a single bubbling scenario, cf. (3.18), which is ruled out in Section 3.4. Hence (3.17) induces a compact flow.
The plan of this work is as follows. In Section 2 we recall some preliminary notions already introduced in [15] for the study of such flows. In particular in Section 2.1 we study the difference or rather the strict similarities of the shadow flow for (1.1) and (1.3), i.e. the dynamics of those variables relevant to the underlying finite dimensional reduction. Subsequently we recall in Section 2.2 some first and easy properties on flow lines based on this reduction. After this lengthy exposition of introduction and preliminary results in Sections 1 and 2 we study in Section 3 all possibilities of non compact flow lines for the flows induced by (1.1) and (1.3) and afterwards of a slight modification of the latter. Precisely we exclude in Section 3.1 all possibilities for non compact flow lines for (1.1) and (1.3), which are not of single bubble type and concentrating at the maximum point of K. Subsequently in Section 3.2 we show, that the latter remaining possibility is realised, i.e. that in fact such non compact flow lines exist for both flows. Finally we modify the latter flows in Section 3.3 and thus introducing a new pseudo gradient flow, which in Section 3.4 is shown to be compact. Last and for the sake of readability we collect in the Appendix 4 some statements from [15] and a proof from Section 2.

Preliminaries
As we had seen via (1.7) and (1.8), every flow line for (1.1) and (1.6) up to the choice of a time sequence constitutes a Palais-Smale sequence for J, whose possible lack of compactness we now describe.
One may expand G a = In addition the positive mass theorem tells, that H a (a) ≥ 0 for all a ∈ M and H a (a) = 0 for M ≃ S n , while H a (a) > 0 for M ≃ S n in the sense of conformal equivalence.
We abbreviate some notation.
and so on.
Let us collect some standard interaction estimates for these bubbles.
For a better description of the gradient we decompose the second variation. To that end we recall from [15], cf. Lemma 3.6 and Proposition 3.7, where {ω, e i : i = 1, . . . , m} ∈ ON B Lg 0 (ker∂ 2 J(ω)) and We call w ∈ U a pseudo critical point related to ω, if We may thereby define a neighbourhood of, where a loss of compactness, if present, has to occur.
We define and call V (ω, p, ε) in case p > 0 a neighbourhood of a potential critical point at infinity.
Note, that u α,β = 0, if ω = 0, and the conditions on α and β k become trivial. Moreover either w = 0 or w > 0 due to the strong maximum principle.
This characterisation of lack of compactness is classical like the subsequent reduction by minimisation and we refer to [4], [15] and [17].
depending on the chosen minimisation. Moreover depend smoothly on u.

The above minimisations evidently induce orthogonal properties for
respectively. This justifies to define the orthogonal spaces, on which v lives.
Recalling Definition 2.2 and u α,β = 0 in case ω = 0 we may simply write depending on the chosen minimisation. These orthogonalities differ only a little, as the next Lemma, whose proof we delay to Appendix 4, quantifies.
The aforegoing Lemma will help us to carry over several estimates from [15], which was based on a representation u = α i ϕ i + v with orthogonalities Proposition 2.10. There exist γ, ε 0 > 0 such, that for any 0 < ε < ε 0 and

This positivity property is well known in either case
and evidently one case follows from the other by virtue of Lemma 2.9. Likewise in case u ∈ V (ω, p, ε), cf. Proposition 5.5 from [15].
Proposition 2.11. There exist γ, ε 0 > 0 such, that for any The invertibility of the second variation on the orthogonal space, on which v lives, then provides a priori estimates.
Proof. The statement for V (p, ε) follows by expanding in v and applying Propositions 2.10 and 4.2. Likewise the statement for V (ω, p, ε) follows by expanding in v and applying Proposition 4.3 and 2.11, where we denote by These estimates on v are upon the appearance of |∂J(u)| instead of |δJ(u)| the same as in [15], cf. Corollaries 4.6 and 5.6 therein. In fact in the latter work we had too graciously estimated against |δJ(u)| in many cases. In what follows we will simply give the correct statements without repeating the various proofs from [15].

The shadow flows
We recall some standard testings of the first variation cf. Proposition 1.1.
So far and in contrast to [15] we have removed the appearance of |δJ|. In fact only in the computation of the shadow flow, i.e. the description of the movements of α i , λ i and a i this error term inevitably enters.
The statements concerning the Yamabe type flow (1.1) are exactly those of Corollaries 4.7,5.7 in [15] and they are proven by testing the flow via ∂ t u, φ l,j . In case of (1.1) the natural scalar product is cf. the proof of Lemma 4.1 in [15].
(ii)  In order to compare (i)-(iii), note, that by virtue of Propositions 4.1 we have cf. Proposition 4.1, also (5.13) in [15] for the analogon in case ω = 0. Consequently and hence, since σ l,j = O(|∂J(u)|) Here enters the difference from (1.1) to (1.3). In fact we have to estimate i.e. there appears |δJ(u)| instead of |∂J(u)|. Also note, that we have along each flow line by virtue of Proposition 2.11 from [15]. We thus obtaiṅ Hence Proposition 2.14 for u ∈ V (p, ε) follows from Proposition 2.13 and (2.1) absorbing v via Proposition 2.12. The case u ∈ V (ω, p, ε) is analogous.

Principal behaviour
Let us recall some generic notions and results in the statements below. Under this mild assumption we have uniformity in V (ω, p, ε) as follows.
Proposition 2. 16. Assume ∂J to be principally lower bounded. For with k u ≡ 1 we then have Proof. Cf. Proposition 6.2 in [15].
As a consequence we obtain limiting uniqueness of non compact flow lines in analogy to the unique limit of compact flow lines.
Proposition 2.17. Assume ∂J to be principally lower bounded. If a sequence u(t k ) along (1.1) or (1.3) diverges in the sense, that then u diverges as well in the sense, that Proof. Cf. Proposition 6.3 from [15] Remark 2.18. In the statement of Proposition 2.17 and in contrast to its corresponding counterpart Proposition 6.3 in [15] we have replaced ". . . converging to a critical point at infinity in the sense, that . . . " by ". . . diverges in the sense, that. . . ".
In fact, as we have exposed in [16] and will see in the present paper, not every non compact or diverging flow line leads to a critical point at infinity.
Note, that Proposition 2.17 in combination with Proposition 2.6 tells us, that every non compact, i.e. diverging flow line has to remain in some V (ω, p, ε) eventually for every ε > 0. Proof. We just have to adapt the corresponding proof of Proposition 6.5 in [15] to this situation. In case ω = 0 Propositions 2.12, 2.13 and (2.1) show To prove (2.4) and (2.5) note, that i =j | up to some o( i =j ε i,j ), whence we immediately obtain (2.5). Plugging (2.4) and (2.5) into (2.3) we obtain for C > 1 sufficiently large In case ∆K i ≥ 0 or |∇K i | > ǫ for ǫ > 0 small we immediately obtain for some c > 0 and all λ i > 0 sufficiently large choosing κ i such, that In particular (2.7) follows in case ∆K i < 0 and |∇K i | = 0, since then by Condition 1.2 Finally in case ∆K i < 0 and 0 = |∇K i | < ε we have and thus by Cauchy-Schwarz inequality Choosing therefore κ i such, that 1 2 n + 2 2 γ 2 < γ 3 κ i and 1 2 n + 2 2 γ 2 < γ 4 κ i , then (2.7) holds true as well and thus in any case. We conclude by definition, the claim follows noticing H i > c > 0 due to M ≃ S n and by means of the positive mass theorem. The case ω > 0 is proven analogously.

Divergence and Compactification
Throughout this section we assume Condition 1.2 to hold true and identify the lack of compactness of the flows on X generated by (1.1) and (1.3). Subsequently will perform a slight variation of these flows and thereby restore compactness.

Compact regions
In   .7) and (1.8), Propositions 2.6 and 2.17 tell us, that we may assume u ∈ V (ω, p, ε) for all times to come for some V (ω, p, ε) and u −→ ω strongly in case ω > 0 and p = 0, in which case u as a flow line is compact.
Hence we may assume, that eventually u ∈ V (ω, p, ε) for ω > 0 and p ≥ 1. Then Proposition 2.14 and the principal lower bound on ∂J, cf. Definition 2.15, give Then ordering 1 λ1 ≥ . . . ≥ 1 λp and recalling (2.4) and Then the right hand side is integrable in time, while necessarily ψ −→ −∞ as some λ i −→ ∞. Hence all λ i have to stay bounded, which due to the principal lower bound on ∂J prevents |∂J(u)| −→ 0, hence contradicting the time integrability of |∂J(u)| 2 .
Lemma 3.2. Every flow line away from x 0 , i.e. eventually Proof. We may assume u ∈ V (p, ε) eventually. Then Proposition 2.14 and the principal lower bound on ∂J, cf. Definition 2.15, give Moreover by assumption Then ordering 1

Recalling (2.4) we then find
since for l i ∈ Q and j ∈ P \ Q by definition hence a li and a j are far from each other and therefore, cf. Lemma 2.3, (1)).
Then clearly Θ ≥ 0 and there holds and hence We then find Due to γ3 γ2 = 3, cf. the proof of Proposition 6.8 in [15], we have 4 · 7γ 2 − 5 · 4γ 3 = −32γ 2 and there holds, cf. (2.6) and arguing as for (2.4), for i > j as we shall prove below. We thus obtain As a consequences Θ, hence all λ i |a i | 5 are bounded and On the other hand for all whence λ i −→ ∞ due to (3.5) necessitates, that for some t k,i k→∞ − −−− → ∞ at least while arguing as before on {λ i |a i | 5 ≥ 2ε} Hence we may assume, that eventually ∀ 1 ≤ i ≤ p : λ i |a i | 5 ≤ 4ε, thus and likewise |ai| 2 So λ m −→ ∞ is impossible and we are left with proving (3.3). Recalling and hence −λ i ∂ λi ε i,j ≥ n−2 4 ε i,j in either of the cases Hence we may assume d g0 (a i , a j ) ≤ 1 λi and λj λi ≫ 1. Since for i > j by assumption we then have |a i | ≫ |a j | and hence d g0 (a i , a j ) ≃ |a i − a j | ≃ |a i |. Therefore However ϑ i = 0 on {λ i |a i | 5 ≤ ε} and we conclude This show the first statement of (3.3). We then compute and observe, that the latter sum is non positive, whence Hence the statement follows for C ≫ 1 sufficiently large, provided we may uniformly bound ϑ j ϑ i for i > j, which recalling (3.2) translates into i.e. monotonicity in case κ = 1. Recalling furthermore η⌊ (0,1) = 0, η⌊ (2,∞) = 1 and η ′ ⌊ (1,2) > 0, evidently (3.6) is satisfied, whenever s > 1 + δ for some δ > 0 small, while we may assume η ′′ ≥ 0 on (0, 1 + δ). Hence ϑ as a sum of products of non negative monotone functions on (0, 1 + δ) is monotone.
Together Lemmata 3.1,3.2 and 3.3 show, that a non compact flow line u has to satisfy u = αδ a,λ + v ∈ V (1, ε) eventually and a −→ x 0 = max M K.

Diverging flow lines
The only possibility left for a non compact flow line of (1.1) or (1.3) under Condition 1.2 is realised.
Proof. We prove the statement under (1.1). The proof under (1.3) is then analogous replacing in particular the appearance of |δJ| by |∂J|. In order to prove, that u remains in V (1, ε) for all times let us define We then have to show T = ∞. We may clearly assume According to Proposition 2.14 and using the principal lower bound on ∂J, cf. Definition 2.15, the relevant evolution equations are where due to k = 1 and hence r k = J(u) we have for some constant κ > 0 during (0, T ) r k = κ(1 + o ε (1)).

Modifying the gradient flow
We finally discuss how to compactify (1.1) and (1.3) in the situation of Lemma 3.4. From Section 3.2 the only critical value for a non compact flow line is Hence it is sufficient to only modify (1.1) and (1.3) on for sufficiently small 0 < ε ≪ δ. To that end consider a cut-off function where | · | denotes the euclidean distance from x 0 in conformal normal coordinates around x 0 . Moreover consider a second cut-off function Hence η V η a η a,λ is well defined on X and We then consider for some C ≥ 1 as a bounded, locally Lipschitz vectorfield on X, which is well defined due to ∇K(a) = −4|a| 2 a = 0 on supp(η a,λ ), and study the flow generated by Clearly k = 1 is preserved as is positivity u > 0 along flow lines and consequently (3.17) induces a flow on X.
for C ≫ 1 sufficiently large, whence we obtain in combination with (1.5) and therefore u exists positively for all times, provided we have uniform a priori bounds on |∂J(u)|, which we derive from Proposition 1.1 using k = 1 and the boundedness of energy along a flow line. The latter boundedness follows from the subsequent Lemma 3.5. Proof. Since ∂J(u)u = 0 by scaling invariance, we clearly have Then Proposition 2.12 and the principal lower bound on ∂J yield cf. Definition 2.15, whence From Proposition 2.13 and (2.1) we then find using again Proposition 2.12 and the principal lower bound on ∂J. Therefore Note, that on supp(η a,λ ) we have λ|a| 2 ≥ ε and hence |∇K(a)| However, since ∂ t u ≤ C under (3.17), as is uniformly bounded along a flow line, and d(∂V (1, 2ε), ∂V (1, ε)) ≥ε, by combining Proposition 2.12 and (i) from Proposition 4.1 with the principal lower bound on ∂J, cf. Definition 2.15. Therefore we infer from Lemma 3.5 which necessitates J(u s k ) −→ −∞, a contradiction. Hence we may assume u ∈ V (1, 2ε) eventually.
On the other hand, since by Lemma 3.5 every flow line up to a sequence in time is a Palais-Smale, cf.
(1.8), we may assume, that u is precompact in some V (ω, p, δ) for every δ > 0. Since 2δ)) >δ in case ω = 0 or p = 1 for all δ > 0 sufficiently small, the same energy consumption argument as before would lead to the same contradiction. Hence necessarily u = αϕ a,λ + v ∈ V (1, δ) for every δ > 0 eventually. (3.18) In particular we may assume η V = 1 eventually for a non compact flow line.
Note, that L g0 φ k,i = c k ϕ generally, cf. Lemma 2.1 in [12]. Hence choosing α suitably, we derive what had to be shown.