Conformal Metrics with Constant Q-Curvature

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.


Introduction
Q-curvature is a scalar quantity introduced by Tom Branson in 1985, [5], which in recent years has attracted lot of attention since it turns out to be fundamental in several questions. Here we will be mainly interested in its role in conformal geometry, and especially in its transformation law under a conformal change of metric.
Let (M, g) be a four dimensional manifold, with Ricci tensor Ric g , scalar curvature R g and Laplace-Beltrami operator ∆ g : the Q-curvature of M is defined by If we perform the conformal change of metricg = e 2w g, for some smooth function w on M , then the Q-curvature transforms in the following way where P g is the Paneitz operator P g (ϕ) = ∆ 2 g ϕ + div g In addition to these, we have an analogy with the classical Gauss-Bonnet formula (χ(Σ) stands here for the Euler characteristic of Σ) Σ K g dV g = 2πχ(Σ).
Indeed, if W g denotes the Weyl tensor of a four manifold (M, g), one has Notice that, from either (1) or (2), it follows that the total Q-curvature of M , which we denote by is a conformal invariant of (M, g): we refer for example to the survey [19] for more details.
Since for two-dimensional surfaces the integral of the Gauss curvature characterizes completely the topology, it is reasonable to expect that conformal invariant objects on four manifolds, like P g or k P , could at least provide some partial geometric information. Indeed some results in this spirit were obtained by Gursky in [34] (see also [33]). If a manifold of non-negative Yamabe class Y (g) satisfies also k P ≥ 0, then ker P g consists only of the constant functions and P g ≥ 0, namely P g is a non-negative operator. If in addition Y (g) > 0, then the first Betti number of M vanishes, unless (M, g) is conformally equivalent to a quotient of S 3 × R. On the other hand, if Y (g) ≥ 0, then k P ≤ 8π 2 with equality holding if and only if (M, g) is conformally equivalent to the standard sphere.
In two dimensions, the uniformization theorem asserts that every compact surface carries a conformal metric with constant Gauss curvature. Such deformation has a strong geometric relevance, since the deformed metric will be a model one: spherical, Euclidean or hyperbolic. In four dimensions one might ask the same question for the Q-curvature: this being a scalar quantity, it will not be sufficient to control the whole curvature tensor, however the new metric will be a privileged one and might have special properties.
Writingg = e 2w g, by (1) the question amounts to finding a solution of the equation where Q is a real constant. Problem (3) is variational, and solutions can be found as critical points of the following functional Here H 2 (M ) is the space of real functions on M which are of class L 2 together with their first and second derivatives, and the symbol P g u, v stands for Notice that the functional II is well defined on H 2 (M ): in fact, by the Adams inequality (see [1,18]) one has log M e 4(u−u) dV g ≤ 1 8π 2 P g u, u + C, where u is the average of u on M and where C depends only on M , so the last term in II is well defined in this Sobolev space. Problem (3) has been solved in [18] for the case in which P g is a non-negative operator and when k P < 8π 2 : by the above-mentioned result of Gursky, sufficient conditions for these assumptions to hold are that Y (g) ≥ 0 and that k P ≥ 0 (and (M, g) is not conformal to the standard sphere). More general sufficient conditions for the above hypotheses to hold have been obtained by Gursky and Viaclovsky in [35].
Under the assumptions in [18], by (4) the functional II is bounded from below and coercive, hence solutions can be found as global minima. The result in [18] has also been extended in [10] to higher-dimensional manifolds (regarding higher-order operators and curvatures) using a geometric flow.
The solvability of (3), under the above hypotheses, is useful in the study of some conformally invariant fully non-linear equations, as shown in [13]. Some remarkable geometric consequences of this study, given in [12,13], are the following. If a manifold of positive Yamabe class satisfies k P > 0, then there exists a conformal metric with positive Ricci tensor, and hence M has finite fundamental group. Furthermore, under the additional quantitative assumption k P > 1 8 M |W g | 2 dV g , M must be diffeomorphic to the standard four-sphere or to the standard projective space.
This note concerns some recent progress about problem (3), in particular an extension of the uniformization result of [18], which is given in [24]. Theorem 1. Suppose ker P g = {const}, and assume that k P = 8kπ 2 for k = 1, 2, . . . . Then (M, g) admits a conformal metric with constant Q-curvature.
The assumptions in Theorem 1 are conformally invariant and generic, and hence apply to a large class of four manifolds, especially to some with negative curvature. For example, products of two negatively-curved surfaces might have total Q-curvature greater than 8π 2 , see [23]. However the theorem does not cover all manifolds, for example conformally flat manifolds with positive and even Euler characteristic, by (2), are excluded.
Our assumptions include those made in [18] and one (or both) of the following two possibilities P g possesses k (counted with multiplicity) negative eigenvalues.
In these cases the functional II is unbounded from below, and hence it is necessary to find critical points which are possibly of saddle type. This is done using a new min-max scheme, which we are going to describe briefly in the next section, depending on k P and the spectrum of P g (in particular on the number of negative eigenvalues k, counted with multiplicity). One fundamental issue in applying variational techniques is compactness: it is possible to prove that indeed solutions of (3) (or of some perturbation) stay compact provided P g has no kernel and k P stay bounded away from 8π 2 N. The following result was proved in [25,37] using blow-up analysis.
Theorem 2 ([37]). Suppose ker P g = {const} and that (u l ) l is a sequence of solutions to satisfying M e 4u l dV g = 1, where k l = M Q l dV g , and where Q l → Q 0 in C 0 (M ). Assume also that Then (u l ) l is bounded in C α (M ) for any α ∈ (0, 1).
We will show below how this result is useful in deriving existence. For reasons of brevity, we will describe the proof of Theorem 1 only: as mentioned, we will find in general solutions as saddle points of II. To use variational techniques we will try to understand the topological properties of the sublevels, in particular the very negative ones.
If k P > 8π 2 , it is easy to see that II is unbounded from below: one can fix a point x ∈ M and consider a test function with the following form Another way to attain large negative value of II is to consider negative eigenvectors of P g with large norm: in the next section we will focus on the first alternative only, and reason looking at how the function e 4u is distributed over different regions of M .
The last section describes other related results and some open problems in this direction.

Sketch of the proof
We give here the main ideas for the proof of Theorem 1: throughout this section we assume for simplicity that P g is positive definite (except on constants), referring to [24] for details when negative eigenvalues are present.

Improved Adams inequality and applications
One of the main ingredients in our proof is an improvement of the Adams inequality on the functions u for which e 4u is spread into multiple regions: here is the precise statement.
where δ 0 is a positive real number, and let γ 0 ∈ 0, 1 ℓ+1 . Then, for anyε > 0 there exists for all the functions u ∈ H 2 (M ) satisfying The original improvement argument is given in [21], where the authors treat H 1 functions on surfaces and the case ℓ = 1: some modifications are needed here to deal with the fourth order operator. The main step of the proof consists in constructing cutoff functions g i which are identically equal to 1 on Ω i and which have mutually disjoint supports. Then one applies (4) to g i u and chooses the index i so that M g 2 i (∆u) 2 dV g is minimal: the remaining terms of M (∆(g i u)) 2 dV g can be treated as perturbations and yield o( u H 2 (M ) ). The above lemma states that if k P < 8(k + 1)π 2 for some natural k, and if u satisfies (7) with ℓ = k, then II(u) stays bounded from below. Since we can choose δ 0 and the Ω i 's arbitrarily, this suggests that if II(u) is sufficiently low, then e 4u has to concentrate near at most k points of M . Using a covering argument this can indeed be made rigorous, yielding the following result. Lemma 2. Assuming P g ≥ 0 and k P ∈ (8kπ 2 , 8(k + 1)π 2 ) with k ≥ 1, the following property holds. For any ε > 0 and any r > 0 there exists a large positive L = L(ε, r) such that for every Lemma 2 basically states that if II(u) is large negative, then the conformal volume e 4u has a shape as in Fig. 1.
If we assume the normalization M e 4u dV g = 1, the figure suggests to consider the following set known in literature as the set of formal barycenters of M . For k = 1 M 1 is homeomorphic to M , but for larger k's this is a stratified set, namely union of open manifolds of different dimensions, whose maximal one is 5k − 1. For further properties of these sets we refer the reader to [3,9]: we are going to need especially the following one, which can be proved with an argument in algebraic topology.

Lemma 3 (well-known). If M is a compact manifold, for any k ≥ 1 the set M k is noncontractible.
Using Lemma 2 one can construct a continuous (and non-trivial) map from very negative sublevels of II into M k . Proposition 1. For k ≥ 1 (see (5)) there exists a large L > 0 and a continuous map Ψ from the sublevel {II ≤ −L} into M k which is topologically non-trivial.
By topologically non-trivial we mean that the action of Ψ * on the homology classes of the sublevel {II ≤ −L} has non-trivial image. The proof of the latter proposition is rather involved and takes some long technical work: we limit ourselves to explain some of the ideas for the construction. It is easy to project when e 4u is close to smooth pieces of M k , but some difficulties may occur when approaching the singularities. In this case, one can project first on the subsets of lowest possible dimension, and then take homotopies with the higher-dimensional ones, see Fig. 2.

The min-max scheme
In order to proceed we need some preliminary notation. For δ > 0 small, consider a smooth non-decreasing cut-off function χ δ : R + → R satisfying the following properties Then, given σ ∈ M k σ = k i=1 t i δ x i and λ > 0, we define the function ϕ λ,σ : M → R as where d i (y) = dist(y, x i ), y ∈ M , with dist(·, ·) denoting the distance function on M . We have then the following result.
It is indeed the last property, together with the non-contractibility of M k , which guarantees the non-triviality of the map Ψ.
We next introduce the variational scheme which provides existence of solutions for (3). Let M k denote the (contractible) topological cone over M k , which can be represented as M k = M k ×[0, 1] with M k × {0} collapsed to a single point. Let first L be so large that Proposition 1 applies with L 4 , and then let λ be so large that Proposition 2 applies for this value of L. Fixing this number λ, we define the following class of maps Π λ = π : M k → H 2 (M ) : π is continuous and π(· × {1}) = ϕ λ,· on M k .
We have then the following properties. Lemma 4. The set Π λ is non-empty and moreover, letting As a consequence, the functional II possesses a Palais-Smale sequence at level Π λ .
To check that Π λ is non empty, it is sufficient to consider the map π(t, σ) = tϕ λ,σ , which is well defined on M k . A Palais-Smale sequence at level c is a sequence (u n ) n such that II(u n ) → c and such that II ′ (u n ) → 0 as n → +∞. To produce it one considers any map in Π λ with sup m∈ d M k II(π(m)) sufficiently close to Π λ and let it evolve via the gradient flow of II, keeping its boundary fixed. By (9) the maximum value of II on the evolved set, which still belongs to Π λ , cannot go below Π λ , so the gradient of II along the evolution has to become small somewhere along the flow, see for example [45].
If Palais-Smale sequences are bounded, then it is easy to see that they have to converge to a solution of (3). Unfortunately boundedness is not known in general, so we tackle this problem using an argument due to M. Struwe, see [44]. For ρ in a neighborhood of 1 we define the functional II ρ : whose critical points give rise to solutions of the equation Running the above variational scheme one finds a min-max value as in Lemma 4, which we call Π ρ , and a corresponding Palais-Smale sequence. One can easily show that Πρ ρ is nonincreasing, and hence differentiable, in [1 − ρ 0 , 1 + ρ 0 ] provided ρ 0 is sufficiently small. Using the following result, which can be proved as in [22], by the above comments and Theorem 2 one deduces then existence of solutions to (3).

the (dense) set of ρ for which the function
Πρ ρ is differentiable. Then for ρ ∈ Λ II ρ possesses a bounded Palais-Smale sequence (u l ) l at level Π ρ .
A particular case of the above variational method has been used in [22] to study a mean field equations on compact surfaces, which in our setting would correspond to the case k = 1.

Final remarks
The set of barycenters of a manifold (or of a set) has been used crucially in literature for the study of problems with lack of compactness, see [2,3]. In particular, for Yamabe-type equations (including the Yamabe equation and several other applications), it has been employed to understand the structure of the critical points at infinity (or asymptotes) of the Euler functional, namely the way compactness is lost through a pseudo-gradient flow. Our use of the set M k , although the map Ψ presents some analogies with the Yamabe case, is of different type since it is employed to reach low energy levels and not to study critical points at infinity. As mentioned above, we consider a projection onto the k-barycenters M k , but starting only from functions in {II ≤ −L}, whose concentration behavior is not as clear as that of the asymptotes for the Yamabe equation.
Theorem 1 has been recently extended to all dimensions in [40]: one of the issues there is that the explicit expression of the Paneitz operator is not known in general, but only the principal part is needed. Also, the blow-up analysis to obtain the counterpart of Theorem 2 relies almost entirely on the Green's representation formula.
The above approach has been also used in [41,42] to study the case of four manifolds with boundary: in fact, in two papers by A. Chang and J. Qing, see [14,15], it was constructed a scalar quantity, the T -curvature, which represents an analogue of the geodesic curvature for surfaces with boundary. If one wants to prescribe constant Q-curvature in the interior and zero T -curvature (and mean curvature) on the boundary, it turns out that the quantization of conformal volume at the boundary is one half of the one in the interior. As a consequence, one needs to modify the topological argument and consider formal barycenters which allow a double number of Dirac deltas on ∂M compared to the interior.
We also mention the papers [11,39,46], where the problem of prescribing non-constant Q-curvature on spheres was considered. In this case the analysis is more delicate since the problem is non compact, due to the presence of the Möbius group. One has then to analyze how compactness is lost and employ topological arguments, like degree of Morse theory, to derive existence results. In fact Morse theory can be used, jointly with the Poincaré-Hopf theorem, to compute the Leray-Schauder degree of (3). By Theorem 2, if P g has no kernel and if k P ∈ 8π 2 N, then the solutions of (3) stay uniformly bounded, so the degree of the equation is well defined, and in [38] it was proven that it is given by where k stands for the number of negative eigenvalues of P g , counted with multiplicity. This result is related to one obtained in [20] for a mean field equation on compact surfaces, and based on sophisticated blow-up techniques: our proof is of different type and relies instead on Morse theory, as mentioned. A natural question one may ask is whether Theorem 1 can be extended to the situations in which P g has non-trivial kernel or when k P is an integer multiple of 8π 2 : in this cases more precise characterizations of P g and Q g (or on their mutual relation) should be given.
One simple situation to consider is when P g has non-trivial kernel and k P = 0. In this case (3) has the following simple expression P g u + 2Q g = 0.
Clearly, by Fredholm's alternative, the equation is solvable if and only if Q g is perpendicular to ker P g . It seems possible to violate this condition but no example so far is known.
If one wishes to study the case of non-trivial kernel in more generality, possibly employing min-max arguments, it would be necessary to understand the behavior of II on ker P g , in particular on functions with large norm. For example, taking v ∈ ker P g , v = 0, one has II(tv) = 4t M QvdV g − k P log M e 4tv dV g .
Since every element in ker P g is smooth we have v ∈ L ∞ (M ), and in the limit t → +∞ one has log M e 4tv dV g ≃ 4t max M v. It follows that for v ∈ ker P g II(tv) = 4t t → +∞.
It would be useful to prove that this function either has constant sign on the unit sphere of ker P g , or that its negative sublevel has non-trivial topology. Under these conditions one might hope to adapt the variational scheme in [24] and prove existence of solutions to (3).
Also the case k p = 8kπ 2 should require some new geometric insight. In the recent paper [36] the authors considered four manifolds such that k p = 8π 2 . They approached the problem perturbing the equation as in (10), letting ρ tending to 1 from below: indeed for ρ < 1 by the Adams inequality II ρ admits a minimizer u ρ . In [36] the asymptotic behavior of u ρ was studied and it was shown that if the function (∆ g,y S(x, y)) y=x + 4|∇ g,y S(x, y) y=x | 2 − R g (x) 8 (11) is positive on M , then u ρ converges to a solution of (3). In the above formula S stands for the regular part of the Green's function of P g : precisely, using the convention that P g G(x, y) + 2Q g (y) = 16π 2 δ x , one has that, near the point x, G satisfies G(x, y) = −2 log dist(x, y) + S(x, y) for some C 2 -smooth S. However, there are so far no known conditions which can guarantee the positivity of the function in (11). A result of this kind would be a counterpart of the positive mass theorem by Schoen and Yau for compact manifolds of positive scalar curvature.