Conformal Classes Realizing the Yamabe Invariant

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(3) More explicitly: let g be a representative of c; and U the set of positive smooth functions u such that (u2g) is a unit-volume Yamabe metrics in c; then there exists a measure µ on U, such that n−2 2 1 2 2 0= u Ric(u g) − n R(u g)u g dµ(u). ZU Corollary 1.2. Let M be a compact smooth n-manifold, and c a conformal class attaining Y (M), and suppose that there are, up to rescaling, only finitely many Yamabe metrics in c. Then there exist N ∈ N, and Yamabe metrics (gi)1≤i≤N (not necessarily unit-volume) in c, such that N 1 1−2/n 0= Ric(gi) − n R(gi)gi |dVolgi | . i=1 X Proof. Write F explicitly as (gj). By Theorem 1.1 there exist nonnegative reals (aj) such that 0 = 1 1−2/n j aj Ric(gj) − n R(gj)gj |dVolgj | . Drop those gj for which aj = 0, and rescale the rest. PThe interest of Theorem 1.1 and Corollary 1.2 lies in their connection to a subtle and very appealing open problem: Question 1.3. Do all conformal classes attaining the Yamabe invariant contain an Einstein metric? This question is motivated by the examples of manifolds for which the Yamabe invariant has to date been computed, which fall into two classes: • Einstein manifolds (M, g) for which it has been proved that [g], the conformal class of the Einstein metric, attains Y (M); these include – Sn ([Aub76b]; see [LP87, Section 3] for an exposition); – manifolds admitting flat metrics ([SY79, GL83]; see [Sch89, Proposition 1.3] for an exposition); – general-type and Calabi-Yau complex surfaces [LeB96, LeB99]; – CP2 [LeB97]; – RP3 [BN04]; • manifolds M for which the existence of a conformal class attaining Y (M) is unclear, or for which it has been proved that no conformal class attains Y (M); these include – S1 × Sn−1 and connect sums thereof [Kob87]; – M#kCP2, where M is a complex surface of general type [LeB96]; – M#[S1 × S3], where M is a 4-manifold with Y (M) ≤ 0 [Pet98]; – RP3#k(S1 × RP2) [BN04]; as well as by the observation that the Yamabe invariant is a minimax quantity associated with the total scalar curvature functional M R(g)dVolg g 7→ 1−2/n , RM dVolg of which Einstein metrics are the critical points. If Question 1.3 were answered in the affirmative, it R could perhaps be possible to find Einstein metrics on new manifolds by direct variational methods, by maximizing the functional I. Theorem 1.1 generalizes two previous results on Question 1.3. In the case when the finite set of Theorem 1.1 consists simply of a single metric g, the following well-known result is obtained: Lemma 1.4 (see for example [Sch89, discussion preceding Lemma 1.2]). Suppose that the conformal class c attains Y (M), and suppose that there is, up to rescaling, exactly one Yamabe metric g in c. Then g is Einstein. 2 1 Proof, given Theorem 1.1. The tensor Ric(g) − n R(g)g vanishes precisely when g is Einstein. In the case when the class c contains exactly two unit-volume Yamabe metrics (i.e., |F| = 2), we may write them explicitly as g and u2g, and obtain the following relationship, previously derived by Anderson [And05, equation 2.37]: −1 n−2 1 −1 1 −1 u (1 + u )[Ric(g) − n R(g)g]= −(n − 2)[Hessg(u ) − n ∆g(u )]. Anderson then gives a Bianchi-identity argument deducing a contradiction from this relationship unless the two metrics coincide, thus answering Question 1.3 in the affirmative in this case: Theorem 1.5 ([And05, Theorem 1.2]). Suppose that the conformal class c attains Y (M), and suppose that there are at most two (modulo rescaling) Yamabe metrics in c. Then in fact there is exactly one Yamabe metric, and that metric is Einstein. We note a different point of view, from which the available evidence regarding Question 1.3 is more equivocal. There is a bidirectional version of Lemma 1.4 which rephrases Question 1.3 as a question of uniqueness: Proposition 1.6 ([Oba72], see also [Sch89, Proposition 1.4] ). Let M be a compact manifold other than the sphere, and suppose the conformal class c attains Y (M). Then there is an Einstein metric in c if and only there is a unique (modulo rescaling) Yamabe metric in c; the Einstein metric is the Yamabe metric if so. In a general (that is, not necessarily maximizing) conformal class, the question of uniqueness (modulo rescaling) of Yamabe metrics is well-studied. Let Ω be the subset of conformal classes containing a unique Yamabe metric. Ω is large (for instance it contains all conformal classes c such that I(c) ≤ 0 [Aub70], and it is open and dense [And05]) but its complement need not actually be empty; the simplest counterexample is the round sphere, and many others have been found [Sch89, BP13].

We therefore hope that future work may use the results of this paper to provide a resolution of Question 1.3, in one of two ways. On the one hand, there may exist a Bianchi-type argument generalizing Anderson’s, which uses Theorem 1.1 to answer the question in the affirmative, at least when F is finite. On the other hand, it may in fact be possible to construct a set of metrics satisfying the algebraic criterion of Theorem 1.1, thus answering the question in the negative. 1.2. The technique. This paper was inspired by two closely analogous results in other geomet- ric minimax contexts. Nadirashvili [Nad96] studies Riemannian metrics g on a compact smooth 2/n manifold M which maximize the weighted first Dirichlet eigenvalue λ1(g)Vol(M, g) , where 2 M |∇u| dVolg λ1(g)= inf . {u∈C∞(M): u2dVol R M g 0= R u} R Among other things, he observes (Theorem 5) that such metrics are λ1-minimal; that is, that there exist a set of first Dirichlet eigenfunctions for g, such that the map of M into Euclidean space defined by those eigenfunctions is an isometric embedding as a minimal submanifold of the unit sphere. The key point of the proof is to find a finite set (ui) of first eigenfunctions, such that 1 g = (du )⊗2 + ∆ (u2)g . i 4 g i Xi Fraser and Schoen [FS12] study Riemannian metrics g on a compact smooth manifold with 1/(n−1) boundary M which maximize the weighted first Steklov eigenvalue σ1(g)Vol(∂M,g|∂M ) , 3 where 2 |∇u| dVolg σ (g)= inf M . 1 0 2 {u∈C (M): ∂M u dVolg|∂M u harmonic, R R∂M u=0} R Among other things, they observe (Proposition 5.2) that such a metric g has a set of first Steklov eigenfunctions, such that the map of M into Euclidean space defined by those eigenfunctions is a conformal embedding as a minimal submanifold of the unit ball, isometric on ∂M. The key point of the proof is to find a finite set (ui) of first eigenfunctions, such that 1 0 = (du )⊗2 − |du |2g , on M, i 2 i g Xi 2 1 = ui , on ∂M. Xi Our argument, particularly in Section 4, closely follows Fraser and Schoen’s. 1.3. Outline. The organization of this paper is as follows. Section 2 reviews some preliminaries. Section 3 is dedicated to the proof of Proposition 3.1; this proposition establishes control on the formal derivative of the Yamabe functional at a conformal class which attains the Yamabe invariant. A Hahn-Banach theorem argument in Section 4 completes the proof of Theorem 1.1. 1.4. Acknowledgements. Discussions with Rod Gover, Richard Schoen and my advisor Gang Tian have helped shape this paper, and I am grateful to them for their interest. I also especially thank Georgios Moschidis, who drew my attention to an error in the main theorem of this paper as it appeared in an earlier version of the paper (see remark following Proposition 4.4).

2. Preliminaries 2.1. Bounds on constant-scalar-curvature metrics. We give a version of the standard bound on metrics with constant scalar curvature, in which the dependence of the bounds on the metric is made explicit. By convention we deﬁne

n n 2/n 2 Γ( )Γ( + 1)ωn−1 Λ := 4n(n − 1) 2 2 = n(n − 1)ω n . Γ(n + 1) n Here ωn is the volume of the n-sphere. In a ﬁxed conformal class, the standard a priori bounds on metrics with constant scalar curvature are essentially equivalent to the solution to the Yamabe problem for (M, [g]) with I([g]) < Λ, and are due to Trudinger [Tru68] and Aubin [Aub76b, Aub76a] (see also the exposition in [LP87]): Theorem 2.1. Suppose given m ≥ 2, p>n, η, and a smooth metric g on M. There exists C > 0 dependent only on n, m, p, η, g, such that for each smooth positive function ϕ on M such that ϕ4/(n−2)g has volume 1 and constant scalar curvature λ ≤ Λ − η, we have the uniform bound

||ϕ||Wm,p,g ≤ C. This theorem holds more generally for a compact set of conformal classes: Theorem 2.2. Suppose given m ≥ 2, p>n, i, D, B, η. There exists C > 0 dependent only on n, m, p, i, D, B, η, such that for each

• smooth metric g on M such that diam(M, g) ≤ D, injrad(M, g) ≥ i, ||Ric(g)||Cm−2 ,g ≤ B, • smooth positive function ϕ on M such that ϕ4/(n−2)g has volume 1 and constant scalar curvature λ ≤ Λ − η, 4 we have the uniform bound ||ϕ||Wm,p,g ≤ C. Such generalizations from a single conformal class to a compact set of conformal classes have generally been understood to be folklore, see for instance [And05, discussion following Equation 2.1], [KMS09, Lemma 10.1]. The point to be checked is the dependence of the constant C in some p standard elliptic estimates on the background Riemannian metric g: the L estimates ||u||Wm+2,p,g ≤ C[||∆gu||Wm,p,g+||u||p,g] (where C can be controlled in terms of the injectivity radius and ||Ric||Cm,g) and the Harnack inequality supM u ≤ C infM u (where C can be controlled in terms of the diameter, the injectivity radius and a lower Ricci bound); in each case one works ﬁrst locally in harmonic co-ordinates [HH97, Heb96] and then globally.

2.2. A compactness theorem for Yamabe metrics. Lemma 2.3. I is upper semi-continuous. Proof. It is the inﬁmum of a continuous functional. In fact I is continuous [BB83], but we will not need that here.

Proposition 2.4. Let m ≥ 3. Let (M, c) be a conformal manifold with I(c) < Λ. Let (ck) be m a sequence of smooth conformal classes on M which C -converges to c, and let (gk) be volume-1 Yamabe metrics for the classes (ck). m−1 Then there exists a subsequence (gki ) which C -converges to a volume-1 Yamabe metric for c.

Proof. Choose p>n, and choose representatives (gk), g of the classes (ck), c with gk → g in the Cm topology. For suﬃciently large k, 1 (1) I(ck) ≤ 2 [I(c) + Λ] (by Lemma 2.3); (2) there are uniform bound on the expressions

diam(M, g ) injrad(M, g ), ||Ric(g )|| m−2 ; k k k C ,gk (3) there exists a uniform C such that for k suﬃciently large, −1 C || · || m+1,p ≤ || · || m+1,p ≤ C|| · || m+1,p . W ,gk W ,g W ,gk 4/(n−2) Therefore, by Theorem 2.2, if ϕk are smooth positive functions on M such that ϕk gk are volume-1 Yamabe metrics for ck, then we have the uniform bound ||ϕk||Wm,p,g ≤ C, hence by Morrey’s inequality the uniform bound ||ϕ || m−1,1− n ≤ C, and so there exists a subsequence (ki) k C p ,g m−1 such that ϕki C -converges. It remains to be checked that the limit, ϕ, makes ϕ4/(n−2)g a Yamabe metric for c. Indeed, 4/(n−2) m−1 4/(n−2) ϕk gk all have volume 1 and constant scalar curvature I(ck). So their C -limit g := ϕ g has volume 1, and constant scalar curvature limk→∞ I(ck), which is ≤ I(c) by Lemma 2.3; this is a contradiction unless equality holds and g is a Yamabe metric.

2.3. Notation. As sketched in the introduction, we denote by |ΛnM| the bundle of densities on M; it is a line bundle, equipped with a natural positive orientation, and hence all real powers |ΛnM|α are well-defined. We write |Ω| for the density associated to an n-form Ω. For each α there is a well-defined “determinant” map of bundle sections, det : C∞(M, Sym2(T ∗M) ⊗ |ΛnM|α) →C∞(M, |ΛnM|2+nα). ⊗2 For α = 0 this is the square of the volume form, g 7→ (dVolg) . The case α = −2/n (that is, the bundle Sym2(T ∗M)⊗|ΛnM|−2/n) has the special feature that its determinant has range C∞(M, R). 5 Denote by M the space of smooth Riemannian metrics on M, and by V the space of smooth positive densities. The map from M to V×C∞(M, Sym2(T ∗M) ⊗ |ΛnM|−2/n) given by −2/n g 7→ (|dVolg|, g ⊗ |dVolg| ) is injective with image V×C, where C denotes the set of smooth positive-definite sections of Sym2(T ∗M) ⊗ |ΛnM|−2/n with determinant 1. The inverse, an isomorphism from V×C to M, is given by (Ω, c) 7→ Ω2/nc. For any element c ∈ C, the set of Riemannian metrics in M corresponding to V × {c} under this isomorphism is precisely a conformal equivalence class of Riemannian metrics. We therefore identify c with that conformal equivalence class, and C with the space of smooth conformal classes. The tangent space TcC to C at c is the kernel of the trace ∞ 2 ∗ n −2/n ∞ trc : C (M, Sym (T M) ⊗ |Λ M| ) →C (M, R). 2.4. Continuity properties of the total scalar curvature functional. We introduce the notation Q : M→ R for the total scalar curvature functional,

R(g)|dVolg| Q(g)= M , 1−2/n R M dVolg and, according to the isomorphism describedR in the previous subsection, have an equivalent functional Q : V×C→ R, 2/n QΩ(c)= Q(Ω c). We have, for a conformal class c,

I(c) = inf Q(g)= inf QΩ(c). g∈c Ω∈V 1 Lemma 2.5. Let Ω be a positive density. The functional QΩ : C → R is C , and its derivative Dc(QΩ) : TcC → R at c ∈C is

2 −1 2/n 1−2/n Dc(QΩ)(w)= Vol(Ω) n hw, −Ric(Ω c)Ω ic. ZM Proof. It is well-known (e.g. [Bes87, Proposition 4.17]) that for a Riemannian metric g and sym- metric 2-tensor h,

R(g)dVolg D (Q)(h) = Vol(g)2/n−1 h, −Ric(g)+ 1 R(g) − M g |dVol |, g 2 Vol(g) g ZM R g ∞ 2 ∗ Recall from the previous subsection that TcC is the set of c-tracefree sections of C (M, Sym (T M)). If w is such a section, the tangent vector (0, w) to V×C at (Ω, c) corresponds to the tangent vector 2/n 2/n 2/n 2/n Ω w to M at Ω c, and since trcw = 0, the term hΩ w, Ω ciΩ2/nc vanishes.

Proposition 2.6 (Modulus of continuity of I). Let (M, c0) be a conformal manifold. Let v be a 2 ∗ n −2/n smooth section of Sym (T M) ⊗ |Λ M| , suﬃciently small that det(c0 + tv) is nonvanishing for t ∈ [0, 1]. Write c0 + tv ct := 1 , det(c0 + tv) n so that (ct) is a 1-parameter family of conformal classes starting at c0. Let Ω be a density such 2/n 2/n that g1 =Ω c1 is a Yamabe metric for c1, and write gt =Ω ct. Then

1− 2 Vol(Ω) n [I(c1) − I(c0)] 1 1 −1/n 1−2/n ≥ v, −Ric(gt)+ R(gt)gt det(c0 + tv) Ω dt. n c Z0 ZM t D 6 E Proof. Since g is a Yamabe metric for c1,

Q|dVolg|(c1)= Q(g)= I(c1), so

I(c1) − I(c0) = Q|dVol |(c1) − inf QΩ(c0) g Ω∈V ≥ Q|dVolg|(c1) − Q|dVolg|(c0). We calculate dc d c + τv τ = 0 dτ dτ 1 τ=t τ=t "det(c0 + τv) n # 1 1 n 1 n −1 v · det(c0 + tv) − (c0 + tv) · n det(c0 + tv) · trc0+tvv det(c0 + tv) = 2 det(c0 + tv) n 1 − n 1 = det(c0 + tv) v − n (trct v)ct , so by Lemma 2.5,

1− 2 d Vol(Ω) n Q (c ) dτ |dVolg| τ τ=t 1 1 − n 1−2/n = hv − n (trct v)ct, −Ric(gt) det(c0 + tv) Ω ict ZM 1 1 − n 1−2/n = hv, (−Ric(gt)+ n R(gt)gt) det(c0 + tv) Ω ict . ZM The result follows by the Fundamental Theorem of Calculus.

3. An “Euler-Lagrange inequality” Let M be a compact connected smooth manifold, and suppose the conformal class c on M attains the Yamabe invariant of the manifold M. Proposition 3.1. For each distributional section v of Sym2(T ∗M) ⊗ |ΛnM|−2/n, there exists a Yamabe metric g for c, such that

1 1−2/n v, Ric(g) − R(g)g |dVolg| ≥ 0. n c ZM D E Proof. If (M, c) is the conformally round sphere, then c contains an Einstein metric g, the round 1 metric; for this g, the tensor Ric(g) − n R(g)g vanishes. The result follows. If (M, c) is not the conformally round sphere, by the solution of the Yamabe problem [Aub76a, Sch84], I(c) < Λ, so the compactness result Proposition 2.4 applies. We now give a proof in these cases using it. 2 ∗ n −2/n Let (vk) be a sequence of smooth c-tracefree sections of Sym (T M)⊗|Λ M| which converge distributionally to v, and let (tk) be a sequence of positive reals such that for all m we have 1 −1 tk||vk||Cm,g → 0. (For instance, one might choose tk := k ||vk||Cm,g.) Thus the sequence

c + tkvk ck := 1 , det(c + tkvk) n ∞ 2/n of smooth conformal classes C -converges to c. Let (Ωk) be densities such that the metrics Ωk ck are volume-1 Yamabe metrics for the classes (ck). 7 Since c attains the Yamabe invariant, we have that for each k,

0 ≥ I(ck) − I(c).

Thus, applying Proposition 2.6 to c and ck, I(c ) − I(c) 0 ≥ k tk 1 1 −1/n 1−2/n ≥ vk, −Ric(gk,τ )+ R(gk,τ )gk,τ det(c0 + τtkv) Ωk dτ, n c Z0 ZM k,τ D E where we deﬁne the 1-parameter families of conformal classes ck,τ by

c + τtkvk ck,τ := 1 det(c + τtkvk) n 2/n and metrics gk,τ by Ωk ck,τ . By the compactness result Proposition 2.4 and a diagonal argument, we may choose a subse- ∞ quence (ki) such that gki converges in C to a volume-1 Yamabe metric, g say, for c, with Ωki ∞ converging in C to |dVolg|. Therefore also the sequence of ﬁelds on [0, 1] × M, (x,τ) 7→ ·, −Ric(g )+ 1 R(g )g det(c + τt v)−1/nΩ1−2/n , ki,τ n ki,τ ki,τ 0 ki ki ck ,τ D E i C∞-converges to the constant-in- τ ﬁeld

1 1−2/n (x,τ) 7→ ·, −Ric(g)+ R(g)g |dVolg| . n c D E By construction, the sequence (vk) of constant-in-τ ﬁelds on [0, 1] × M converges distributionally to the constant-in-τ ﬁeld v. Pairing, it follows that

1 1−2/n v, −Ric(g)+ R(g)g |dVolg| n c ZM D 1 E 1 −1/n 1−2/n = lim vki , −Ric(gki,τ )+ R(gki,τ )gki,τ det(c0 + τtki v) Ωk dτ i→∞ n i c Z0 ZM ki,τ ≤ 0. D E

4. Hahn-Banach argument 4.1. Preliminaries from functional analysis. Theorem 4.1 (Hahn-Banach separation theorem [Rud91, Theorem 3.4], [Edw95, Corollary 2.2.3]). Let X be a locally convex topological vector space over R, and A and B nonempty closed convex subsets of X, with B compact. If for all functionals ϕ ∈ X∗ we have sup ϕ(a) ≥ inf ϕ(b), a∈A b∈B then A ∩ B 6= ∅. Theorem 4.2 ([Phe01, Proposition 1.2], [Edw95, Corollary 8.13.3]). Let Y be a Hausdorﬀ locally convex topological vector space over R, and K ⊆ Y a nonempty compact subset. If a point y0 ∈ Y is in the closed convex hull of K, then there exists a Borel probability measure µ on K, such that

y0 = y dµ(y). ZK 8 Remarks. (i) The integral is to be taken as a Pettis integral ([AB06, Section 11.10], [Edw95, ∗ Section 8.14]). That is, for all functionals ψ ∈ Y , ψ(y0)= K ψ(y) dµ(y). (ii) By a straightforward argument, the converse is also true. R Proof. Since the closed convex hull of K contains y0, there exist sequences Nk of natural numbers, k k (aγ )1≤γ≤Nk of finite sets of positive reals, and (yγ )1≤γ≤Nk of finite sets of elements of K, such that Nk k Nk k k 1 = γ=1 aγ for all k and y0 = limk→∞ γ=1 aγ yγ . Define a sequence of discretely-supported Nk k ∗ probability measures (µk) on K by µk := a δ k . Thus for all functionals ψ ∈ Y , ψ(y0) = P Pγ=1 γ yγ lim ψ(y)dµ (y). k→∞ K k P Applying the Banach-Alaoglu theorem to to the set of Radon measures on K, there exists a R subsequence (µki ) which weak-*-converges to some Borel probability measure µ on K. For all ∗ functionals ψ ∈ Y , ψ(y0) = limi→∞ K ψ(y)dµki (y)= K ψ(y) dµ(y).

4.2. Proof of Theorem 1.1. Let MR be a compact smoothR n-manifold, and c a conformal equivalence class on M. Lemma 4.3. The conformal equivalence class c induces canonical pairings h·, ·i : Γ(Sym2(T ∗M) ⊗ |ΛnM|s) ×C∞(Sym2(T ∗M) ⊗ |ΛnM|1−4/n−s) → R, (Γ denoting distributional sections) such that the induced maps Γ(Sym2(T ∗M) ⊗ |ΛnM|s) → (C∞(Sym2(T ∗M) ⊗ |ΛnM|1−4/n−s))∗ are isomorphisms of topological vector spaces.

4/n Proof. Pick a metric g ∈ c. The induced pairing hv, wi = g(v, w) |dVolg| is easily checked to be independent of the choice of g ∈ c. R Suppose henceforth that the conformal class c attains Y (M). Denote by F⊆C∞(Sym2(T ∗M)) the set of volume-1 Yamabe metrics in the conformal class c. Define a function Q : c →C∞(Sym2(T ∗M) ⊗ |ΛnM|1−2/n) on the set of metrics in the conformal class c, by 1 Q(g) := Ric(g) − R(g)g |dVol |1−2/n. n g Proposition 4.4. The closed convex hull of the set Q(F) contains 0. Proof. Let A be the closed convex hull of the set Q(F). The topological vector space of sections C∞(Sym2(T ∗M) ⊗ |ΛnM|1−2/n) is Fréchet and therefore locally convex. Thus it suffices to verify the hypothesis of Theorem 4.1. Indeed, let v be in Γ(Sym2(T ∗M) ⊗ |ΛnM|−2/n) =∼ (C∞(Sym2(T ∗M) ⊗ |ΛnM|1−2/n))∗ (the isomorphism is by Lemma 4.3). By Proposition 3.1, there exists (rescaling if necessary) a metric g ∈ F such that hv, Q(g)i≥ 0. Since Q(g) ∈ A, it follows that indeed supa∈Ahv, ai≥ 0= hv, 0i. The set F, the set of volume-1 Yamabe metrics in the conformal class c, is compact by Proposition 2.4. So Q(F) is also compact. 9 Remark. An earlier version of this paper featured the statement that the convex hull of Q(F) is closed – and therefore that the convex hull of Q(F) itself contains 0, with a corresponding alteration in the paper’s main theorem. It was claimed that the closedness of the convex hull of Q(F) follows from the compactness of Q(F). This argument is incorrect; see for example [AB06, Example 5.34]. We do not know whether the convex hull of Q(F) must be closed.

Proposition 4.5. There exists a Borel probability measure µ on Q(F), such that 0= Q(F) a dµ(a). Proof. Combine Proposition 4.4 with Theorem 4.2. R

Theorem 1.1 now follows by choosing an arbitrary Borel measure µ on F such that Q∗µ = µ.

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