On Conformal Metrics with Constant Q-Curvature

Andrea Malchiodi

Scuola Normale Superiore,Piazza dei Cavalieri 7,50126 Pisa,Italy

Received 30 July 2018;Accepted(in revised version)12 August 2018

Abstract. We review some recent results in the literature concerning existence of conformal metrics with constant Q-curvature. The problem is rather similar to the classical Yamabe problem:however it is characterized by a fourth-order operator that might lack in general a maximum principle. For several years existence of geometrically admissible solutions was known only in particular cases.Recently,there has been instead progress in this direction for some general classes of conformal metrics.

Key Words: Geometric PDEs,variational methods,min-max schemes.

1 Introduction

A classical problem in conformal geometry is the Yamabe problem,consisting in deforming a background metric on a compact manifold(M,g) so that its scalar curvature becomes constant. This can be considered as an extension of the classical uniformization problem for two-dimensional surfaces and has received a lot of attention in the literature,see[34]for a general introduction to the problem.

The scalar curvature of a manifold transforms conformally according to the law

where Lgis the conformal laplacian,defined by

The latter operator transforms covariantly,namely one has The Yamabe problem then amounts to finding a positive solution to(1.1)with R ˜g equal to a constant. This constant can be viewed as a Lagrange multiplier when considering the following minimization problem

It can be proved using(1.1)and(1.2)that the latter quantity is conformally invariant.

We next discuss some higher-order analogue of the above problem. In[2]T.Branson introduced the following fourth-order operator in dimension n≥5:

where

The function

Moreover one has the following conformal transformation law

analogous to(1.1).

Formulas(1.4)and(1.5)naturally suggest a higher order version of the Yamabe problem,i.e.,given(Mn,g)does there exists a conformal metric of constant Q-curvature?Due to(1.5),this amounts to finding a positive solution of

with λ∈R.

In dimension four the corresponding equation is

with conformal metric written as ˜g=e2wg. In this case the conformal factor is always positive and the progress on this problem was obtained earlier compared to the higherdimensional case,see[3,8,10,23,33,36]. For prescribed variable curvature,we also mention the papers[7,15,21,37].

In dimensions n≥5,there were some results in particular cases. In[9]some compactness results for solutions of(1.6)were proven,assuming that that Pgwith constant coefficients(as it happens for Einstein metrics),which allows to factorize the Paneitz operator as product of monomials in the Laplacian and to apply the maximum principle iteratively(see also[42]). In[13],for manifolds of dimension n≥8 and non locally-conformally flat(as a counterpart of[1]),solutions to the following equation were found

However the authors were not able to determine the sign of solutions, so these could have been possibly non geometrically admissible.

In[38]the authors were able to find positive solutions to(1.6)on locally conformally flat manifolds of positive scalar curvature. This was done using the approach in[40]via the developing map, under the assumption that the Poincar´e exponent of the manifold is less than (n-4)/2. In [25] the authors proved a compactness result for (1.6) on locally conformally flat manifolds with positive scalar curvature,assuming positivity of the Paneitz operator and of its Green function.They also needed a suitable version of the positive mass theorem for the Green’s function of the Paneitz operator,verified later in[26].Other results about the prescription of Q-curvature can be found in[4-6,11,12,20,24,31].

In this note we are going to describe some recent progress on(1.6),from papers where suitable maximum principles were proved for the Paneitz operator. In[19],the following pointwise conditions were imposed on the Q-curvature and the scalar curvature

Under these assumptions it was proved that the Paneitz operator enjoys a maximum principle,namely that if a smooth function u satisfies Pgu≥0,then either u＞0 or u≡0 on Mn. This result was proved by showing, naively, non-negativity of the scalar curvature and then for the conformal factor(the rigorous proof using a homotopy argument).

As an extension of a result from [22] (concerning the four-dimensional case), under the same assumptions it was also proved that the Paneitz operator is positive-definite.As a consequence of this fact one has that the Green’s function with pole at p, denoted GP(p,·), exist and it is positive away from p. In fact, something more precise can be deduced: if n≤7 or if(M,g)is locally conformally flat,it can be proved that GPsatisfies

with α ≥0 and α=0 if and only if the manifold is conformally equivalent to the round sphere,a fourth-order version of the positive mass theorem.

Consider then the quantity

When M is not conformal to the round sphere and either n=5,6,7 or M is locally conformally flat,the above property was used to prove that there exist conformal factors for which Q is non-negative (and not identically zero), the scalar curvature is positive and such that Fg0[u]＜Sn,the spherical Sobolev constant(see(4.1)). Exploiting a result in[13],the same was shown for dimension n≥8 still for non locally conformally flat manifolds.In [19] it was then introduced a parabolic flow that preserves positivity of the conformal factor and of the scalar curvature and that sequentially converges to a(geometrically admissible)solution of(1.6).

After[19]appeared,in other papers the above results were made progressively conformally invariant. In[28]a maximum principle was proved for the Paneitz operator of manifolds with positive Yamabe invariant and non-negative(and non-zero)Q-curvature.Under the same assumptions,problem(1.6)was attacked in[27]using a dual formulation of Fg0. In particular,the authors considered the following quantity

They showed that if(M,g)is not conformally equivalent to the round sphere,then Θ4(g)is strictly larger than Θ4(gSn)and that maximizers exist and lead to solutions of(1.6)(in relation to a result in[1]). Finally,in[16]the following quantity was introduced:

It was shown via a continuity argument that if n ≥6, Y(g)＞0 and Y*4(g)＞0, then it is possible to find a metric conformal to g with positive scalar curvature and positive Qcurvature. By the results in [19], this implies the existence of a conformal metric with constant and positive Q-curvature.

The restriction on dimension is believed to be just of technical nature and it is expected that it could be removed. It would also be interesting to find such kinds of uniformization results in cases when either the Yamabe quotient or the Q-curvature (or both) are negative. Another interesting aspect is to analyse the compactness of solutions of (1.6):apart from those in[25],some results in this direction are available in[43]and[35].

The plan of the paper is the following. In Section 2 we describe the maximum principle proved in [19], as well as the extension of the positive mass theorem in [26] to arbitrary manifolds not globally conformal to the round sphere. In Section 3 we describe a conformal flow introduced in[19]that preserves the positivity of the conformal factor and satisfies suitable monotonicity properties. In Section 4 we prove sequential sequence of this flow, once we choose suitable initial data with low Sobolev quotient. Finally in Section 5 we describe some conformally (or partially-conformally) invariant extensions that provide existence of metrics with positive and constant Q-curvature.

2 Maximum principle for the Paneitz operator and its Green’s function

In this section we discuss a maximum principle for the Paneitz operator and study some relevant properties of its Green’s function.

Before discussing the maximum principle,we state a preliminary lemma relating the Q-curvature and the scalar curvature via an iteration of the classical maximum principle.

Lemma 2.1. Suppose(Mn,g)is a compact manifold,with n≥5.Assume that Qg≥0 with Qg＞0 somewhere and that Rg≥0. Then we have the strict inequality Rg＞0.

Proof. By definition of Qg,the Q-curvature can be written as

with c1(n),c2(n)＞0. Since Qgis non-negative,we have that

By the strong maximum principle,it follows that either Rg＞0 or Rg≡0. In the latter case,(2.1)would imply

a contradiction.

We have next the main result of this section.

Theorem 2.1. Suppose(Mn,g)is a compact manifold,with n≥5. Assume also that Qg≥0 with Qg＞0 and that moreover Rg≥0. If u∈C4satisfies

Then u0≡1 and u1≡u. Suppose that minMnu≤0,and define λ0∈(0,1]by

Then if 0＜λ＜λ0we must have uλ＞0. Define also the metrics

and let Qλbe the Q-curvature of gλ. Note also that for 0 ＜λ ＜λ0, we have Qλ≥0 and Qλ＞0 somewhere. This is a consequence of the following formula

Since λ＜λ0≤1,by the assumptions on Qg,we have that Qλ≥0 and it is not identically zero.

Let now Rλdenote the scalar curvature of gλ. We will show that if 0 ≤λ ＜λ0, then Rλ＞0. This is indeed true if λ=0. Assume now by contradiction that there exists λ1∈(0,λ0)with minRλ1=0: this would however contradict Lemma 2.1.

By(1.1)it follows that

Since Rλ＞0,the conformal factor uλsatisfies the differential inequality

Since Qgis somewhere positive, this contradicts the fact that Pgu ≥0. We conclude that either u≡0 or u＞0.

We next state a result on the positivity of the Paneitz operator,proved in[19].We only report the proof of the last statement,for brevity reasons and limit ourselves to mention that the arguments missing here rely on a Bochner formula and a subtle integration by parts.

Proposition 2.1. Under the assumptions of Theorem 2.1 the first eigenvalue of the Paneitz operator is positive and hence Pgis invertible. As a consequence, we also have the following inequality for a Sobolev-type quotient

Moreover,if GPdenotes the Green’s function of the Paneitz operator with pole at p∈Mn,then GP＞0 on Mn{p}.

Proof. We just prove the latter assertion.Let(fj)be a sequence of non-negative functions,whose supports shrink to{p},p∈M and such that∫M fjdµg=1 for all j.Then fj⇀δpin the sense of distributions. By the property on eigenvalues of Pg,we have a unique solution Gjto PgGj=fj.By standard regularity theory one has that

By Theorem 2.1 it follows that Gjis positive on M,and hence GP(p,·)≥0 on Mn{p}.

From the proof of Lemma 2.1(see(2.10)),it follows that Gjsatisfies

Therefore we have that

By the strong maximum principle,GP(p,x0)=0 implies GP≡0,a contradiction.

We next study in more detail the regularity of the Green’s function GPwith pole at p.Recall the construction of conformal normal coordinates constructed in[34]: first,a suitable choice of conformal factor is made and then standard normal coordinates are used for this deformed metric. The following result is proved via the classical method of parametrix,with a careful expansion of the lower-order terms in the Paneitz operator.

Proposition 2.2. Let (Mn,g) be as in Theorem 2.1. Suppose also that either n=5,6, or 7,or that n ≥5 and that(Mn,g) is locally conformally flat. For p ∈M,consider the with conformal metric ˜g as in the construction of conformal normal coordinates. Then there exists a constant α such that in those coordinates one has

for 1≤j≤k,with r=|x|=d˜g(x,p).

We next prove a positive-mass theorem for the Paneitz operator,which will be useful for finding conformal metrics with constant Q-curvature. The proof is an adaptation of the argument in[26],where the result is obtained in the locally conformally flat case.

Theorem 2.2. Suppose the conditions of Theorem 2.1 hold true and let α be the constant given in Proposition 2.2. Then one has α≥0,with α=0 if and only if(Mn,g)is(globally)conformally equivalent to the round sphere.

By(1.4)we have

where

is the Schouten tensor. Recalling the definition of the Paneitz operator,we also have

For a small δ＞0,let Bδbe the geodesic ball centered at p of radius δ＞0,with respect to the metric g. Integrating(2.18)on MnBδand applying the Green’s formula we get

where ν is the outer unit normal to ∂Bδwith respect to the metric.

Using(1.2)we find also

Let r(x)=dg(x,p)denote the distance from p with respect to the metric g. By Lemma 6.4 of[34],one has the expansion

Together with Proposition 2.2,we then have

From the asymptotics of the Green’s functions and the fact that, in conformal normal coordinates,Rg=O(r2)we find

By(2.22),

and therefore

It is easy to check that

hence from(2.23)and(2.26)we deduce

Concerning the surface measure,we have the transformation law

Therefore,the first boundary term in formula(2.19)becomes

while the second one satisfies

These imply

As a consequence, we have α ≥0. If α=0, thenhas vanishing Ricci curvature and hence(Xn,)is isometric to the Euclidean space(see[39],page 492). The proof is thereby concluded.

3 A conformal flow

Throughout this section we will always assume that (M,g0) is a compact manifold satisfying the assumptions of Theorem 2.1. By Proposition 2.1, the Paneitz operator Pg0is invertible and hence it makes sense to consider the initial value problem

It is rather standard,via a fixed point argument,to show the following result.

Lemma 3.1. There exists T∈(0,+∞]such that the flow(3.1)has a smooth solution for 0≤t＜T.

We first show that the positivity of the conformal factor is preserved.

Proposition 3.1. For every 0≤t＜T one has the inequality

Proof. Formula(3.1)implies

which in turn leads to

Integrating this inequality we obtain

This implies Pg0u≥0 with Pg0u＞0 somewhere. The strong maximum principle in Theorem 2.1 implies u＞0 for t∈[0,T),giving us the desired assertion.

Remark 3.1. As a consequence of the latter proof one has that Qg＞0 for all t ∈(0,T).Since u is positive,(3.3)implies that

from which we deduce that Pg0u＞0 for every t∈(0,T).

Since u＞0 for all existence times,(3.1)can also be rewritten as

We show next that along the flow we have a conserved and a monotonic quantity.

Lemma 3.2. If u solves(3.6),then for 0≤t＜T we have that

and for the total volume the inequality

where

In particular,we have that

Finally,we have the following upper bound

Proof. (3.6)and the definition ofµimply that

which yields the first statement. To prove the second we notice that

We also have

From(3.12)and(3.13),we deduce(3.8).

The upper bound on the conformal volume follows from the fact that the Paneitz-Sobolev constant is positive:

so V ≤C(g0). This concludes the proof.

Corollary 3.1. One has the estimate

Proof. From the upper bound on volume it follows that

On the other hand,from the positivity of Pg0one finds

so the first inequality in (3.14) follows. The second one is a consequence of the lower bound on the Paneitz-Sobolev quotient.

We next check the long-time existence of the above flow.

Proposition 3.2. The flow (3.1) admits a global in time solution. Moreover, given an initial datum g0there exist positive constants C,C′such that

Proof. Fix a number s＞1: since u＞0 and Pg0u＞0 for t∈[0,T),by(3.3)one has

For the second integral in the last line,by H¨older’s inequality

Choose now s such that

One can apply H¨older’s inequality once more to find

By the Sobolev embedding theorem and by the fact that Pg0＞0 we have

It follows that

for s as in(3.19). Substituting this into(3.20)and using the conformal volume bound of Lemma 3.2 we have

Inserting this estimate into(3.17)gives

Integrating in time one finds

By the Sobolev embedding,one has also

If then s is sufficiently close to n/4,we deduce that

for any p＞1.

Take now s=n/4+1. Returning to(3.18),we have that

Inserting this inequality into(3.17)gives

Integrating in time and using Sobolev’s embeddings we conclude that

for some α ∈(0,1). This implies (3.16). By (3.3), one has also that the Cα-norm of Pg0u grows at most exponentially fast. Therefore the C4,α-norm of u also grows at most exponentially fast,so we must have global existence in time.

4 Convergence of the flow

In this section we describe the converge of the above flow once suitable initial data are chosen. We want in particular to obtain initial conformal data so that they satisfy the assumptions of Theorem 2.1 and so that the Paneitz-Sobolev quotient is low enough. The estimate of the latter one depends on the dimension and on the local conformal flatness and is stated separately in the next two propositions.

Proposition 4.1. Let (Mn,) be a compact manifold of dimension n ≥8. Suppose that Q¯g≥0,Q¯g/≡0,that R¯g≥0 and that(Mn,)is not locally conformally flat.

If the Weyl tensor W(x0)at x0∈M does not vanish,then for ε＞0 small there exists a positive function ψε∈C∞such that

and

where cnis a dimensional constant and where Snis defined by

We give next some ideas of the proof. Esposito and Robert considered in [13] the following test function,

where η(x) is a cut-off supported in B2δ(x0) and identically equal to 1 in Bδ(x0). In the same paper it was shown that for ε＞0 small one has the inequalities

and

These allowed to show that the infimum of F is achieved if n ≥8 and M is not locally conformally flat. However the positivity of a minimizing conformal factor was not guaranteed and therefore it could give rise to a solution not geometrically admissible.

In[19]the latter test function was modified in order to achieve some sign condition on the conformal factor, the scalar curvature and the Q-curvature. Recalling the invertibility of Pgfrom Proposition 2.1(and its conformal covariance),the function ˆuεwas then defined by

By Theorem 2.1, the conformal metric induced byhas the desired sign properties,including the curvatures. In[19]the difference betweenandwas also estimated and it was shown that

giving the desired result.

We have then a related result for low dimensions or in the locally conformally flat case.

Proposition 4.2. Let(Mn,)be a compact manifold of dimension n,with n=5,6,or 7;or letbe locally conformally flat of dimension n≥5. Suppose thatand that. Ifis not conformally equivalent to the round sphere, then for ∈＞0 small and x0∈M,there exists a positive ψε ∈C∞and a constant cx0＞0 such that

To prove the result, consider a cut-off functionequal to 1 in B1and equal to zero outside B2. Define thenand the functiondefined by

The functional F can be well estimated on ˇuεvia an integration by parts, since the Paneitz operator vanishes identically on the Green’s function, away from the point x0.Using this fact and Lemma 4.1 one can show the following estimate.

This one, with a scaling argument used to estimate the latter integrals, leads to the inequality(4.4).

We next show convergence of the flow, under the assumptions of Proposition 4.1 or 4.2,to a solution of the constant Q-curvature equation.

Theorem 4.1. Let(Mn,¯g)be a compact manifold of dimension n≥5 not conformally equivalent to the standard sphere. Suppose also that Q¯g≥0,Q¯g/≡0and that R¯g≥0.

Let g0=h,where h is the metric constructed in Proposition 4.2 or Proposition 4.1. Then the flow(3.1)has a solution for all time and satisfies

for some constant C0＞0. Moreover,there exists tj↗∞such that uj=uj(tj,·)weakly converges in W2,2(Mn,g)to a solution u＞0 of

Proof. Take as initial metric g0be the one given by Proposition 4.1 or 4.2. Proposition 3.2 then implies that(3.1)is defined for all times. Moreover,we have that

with u0≡1 and where ∈0is some positive constant. Lemma 3.2 then implies

for all times.

Recalling(4.1), on(M,g0), given δ ＞0, we can use a standard localization argument to prove

which in turn implies

Inserting(4.8)into(4.9)gives

Taking δ=∈0/10, the first term on the right-hand side can be brought to the left-hand side,so we deduce

Notice that the left-hand side is a power of the conformal volume,so we conclude

for all times,as desired.

By Lemma 3.2 and Corollary 3.1 there exist tj↗∞such that uj=u(tj,·)andµj=µ(tj)satisfy

Therefore u≥0 satisfies

hence u is a strong solution of

By Theorem 2.1,it follows that u＞0,concluding the proof.

5 Conformally invariant extensions

We next mention some results from[27,28]and[17],where some conformally-invariant extensions of the above results were found.

We start with some results in[28](discussed here only for n≥5),where the pointwise positivity of the scalar curvature is replaced by the one of the Yamabe invariant.

The proof of this result relies on(1.4),which implies

Lemma 5.1(see[28]). Let(Mn,g)be as in Proposition 5.1 and let u:M→R be a smooth function such that u≥0 and Pgu≥0. If u vanishes somewhere on M,then u≡0.

Proof. By(5.1)we have that

The same argument gives the conclusion assuming only that u is of class L1,smooth near p and that the inequality Pgu≥0 is satisfied in the distributional sense.

The above result yields global positivity of the conformal factor.

Proposition 5.2(see[28]). Let(Mn,g)be compact,with n≥5,Y(g)＞0 and Qg≥0.Suppose that u:M→R is smooth,satisfies Pgu≥0 and it is not identically constant. Then u＞0 on M.

Proof. Arguing by contradiction,suppose that-λ=u(p)=minMu≤0. This implies that u+λ≥0,u(p)+λ=0 and that Pg(u+λ)≥λQg≥0. Lemma 5.1 implies that u≡-λ,which is impossible.

An easy consequence of this fact is that if Y(g)＞0 and Q ≥0, then the kernel of the Paneitz operator only consists of constant functions and that this kernel is identically zero if Q is also not identically zero.

Lemma 5.2. Suppose(Mn,g)is such that n≥5,Y(g)＞0,Qg≥0 and it is not identically zero.Then kerPg=0 and GP(p,q)＞0 for all q/=p.

Proof. We only need to prove the latter property. Given a smooth function f on M,there exists a unique solution u of Pgu=f,which can be written as

If f is non-negative,from Proposition 5.2 it follows that u ≥0 and therefore GP(p,q)≥0.If GP(p,q)vanishes at some point q,the observation at the end of the proof of Lemma 5.1 implies GP(p,·)≡0,which is a contradiction.

We can now state one of the main results in[28].

Theorem 5.1(see[28]). Suppose(Mn,g)satisfies n ≥5 and Y(g)＞0. Then the following are equivalent

(a) there exists a smooth positive function ρ on M such that Qρ2g＞0,

(b) kerPg=0 and GP(p,q)＞0 for all p/=q,

(c) kerPg=0 and there exists p∈M such that GP(p,q)＞0 for all q/=p.

Proof. The first property implies the second by Lemma 5.2, the conformal invariance of kerPgand the fact that

The second property implies the first by the Krein-Rutman theorem.

Clearly,the second property implies the third one. For the opposite implication,suppose there exists p0such that GP(p0,q)＞0 for all q/=p0. Setting

we have Θ(p0)＞0. We must also have Θ(p)/=0 for all p ∈M,from the latter comments in the proof of Lemma 5.2. By connectedness of M, it must be Θ(p)＞0 for all p ∈M,concluding the proof.

The above results allow to prove a conformally-invariant counterpart of the existence result in Theorem 4.1. In [27] the following quantity was introduced, motivated by a duality approach.

where GPf stands for the convolution

With arguments somehow related to those for the proof of Proposition 4.1,the following result was proved.

Proposition 5.3. Suppose(Mn,g)is a compact manifold with n ≥5,Y(g)＞0 and Q ≥0,Q/≡0. Then Θ4(g)≥Θ4(gSn),with equality satisfied if and only if(Mn,g)is conformally equivalent to the round Sn.

We do not report the proof here for reasons of brevity.The latter result allows to prove convergence of maximizing sequences, by showing that these cannot develop blow-up points.

The following weak convergences also hold true

It is possible to show that

where

Therefore we obtain

We have now the following result.

Theorem 5.2. Suppose(Mn,g) is a compact manifold with n ≥5,Y(g)＞0 and Q ≥0, Q/≡0.Then one has the following properties.

(a) Θ4(g)≥Θ4(Sn),with equality if and only if(Mn,g)is conformally equivalent to the round sphere Sn.

We next describe some results from [17], in which the sign assumption on the Qcurvature is replaced by a conformally covariant condition. We introduce the following three quantities

We clearly have the inequalities

The following result was proved in[17].

Here are two consequences of the above result.

Corollary 5.1. Let(Mn,g) be a compact Riemannian manifold with n ≥6. Then the following are equivalent.

i) Y(g)＞0 and Pg＞0,

We do not give a complete proof of Theorem 5.3 here for reasons of brevity: we just limit ourselves to describe the main strategy used in [17], which relies on a homotopy argument.

Recall that the Schouten tensor is defined as

and its σ2-curvature as

Given t≥1,the authors in[17]considered the following functional:

Critical points of this functional,restricted to metrics with unit volume,are solutions of the problem:

Define then the function f by

Fixing this function f,one can then consider the following problem

Define then the set

Clearly t0∈S: it is then proved via an implicit function theorem that S is relatively open in[1,t0]and via some a-priori estimates that it is also relatively closed,showing that also 1∈S and giving the desired conclusion.

Acknowledgements

The author has been supported by the project Geometric Variational Problems and Finanziamento a supporto della ricerca di base from Scuola Normale Superiore and by the grant from MIUR Bando PRIN 2015 2015KB9WPT001. He is also member of GNAMPA as part of INdAM.