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S^1-equivariant Yamabe invariant of 3-manifolds

We show that the S^1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S^1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S^1-action. Furthermore, we prove a convergence result for the equivariant Yamabe constants of an accumulating sequence of subgroups of a compact Lie group acting on a closed manifold.


Overview over the classical Yamabe invariant
Aubin [9] proved that the Yamabe constant of (M, In dimension 2, the Yamabe invariant is a multiple of the Euler characteristic. For n ≥ 3 it is in general a difficult problem to compute the Yamabe invariant, and only in few cases it can be calculated explicitly. Aubin [9] proved for the ndimensional sphere σ(S n ) = µ(S n , [g st ]) = n(n−1)(vol(S n , g st )) 2/n . Kobayashi [18] and Schoen [29] proved that σ(S n−1 × S 1 ) = σ(S n ). For many closed manifolds M , one can show σ(M ) = 0 as the existence of metrics with positive scalar curvature is obstructed, whereas conformal classes [g i ] with µ(M, [g i ]) → 0 can be written down explicitly. For example the n-torus T n does not carry a metric of positive scalar curvature which can be shown with enlargeability type index obstructions by Gromov and Lawson or with the hypersurface obstruction by Schoen and Yau. For the standard metric g 0 we have µ(T n , [g 0 ]) = 0, so σ(T n ) = 0. Similarly we know σ(M ) = 0 for all nilmanifolds, and quotients thereof.
In order to determine non-zero values for σ, many modern techniques were used: Ricci-flow, Atiyah-Singer index theorem, Seiberg-Witten theory, and the Bray-Huisken inverse mean curvature proof of the Penrose inequality. In dimension 3, values for the Yamabe invariant of irreducible manifolds were already conjectured and partially studied in [6,7].
For example, on a hyperbolic 3-manifold H 3 /Γ the supremum in the definition of the Yamabe invariant σ(M ) is attained in the conformal class of the hyperbolic metric g hyp , and the infimum in the definition of µ(H 3 /Γ, [g hyp ] is attained in g hyp . More generally, it follows from Perelman's work on the Ricci flow that for 3-manifolds with σ(M ) ≤ 0, the value of σ(M ) is determined by the volume of the hyperblic pieces in the Thurston decomposition. We learned this from [17,Prop. 93.10 on page 2832], but ideas for this application go back to [8]. In the case σ(M ) > 0, n = 3, M is the connected sum of copies of quotients S 2 × S 1 and of quotients of S 3 . For connected sums of copies of S 2 ×S 1 we have σ(M ) = σ(S 3 ) but the precise value cannot be determined in most cases. Using inverse mean curvature flow, the Yamabe invariants of RP 3 and some related spaces were determined in [13] and [1], e.g. σ(RP 3 ) = 2 −2/3 σ(S 3 ). This is indeed a special case of Schoen's conjecture explained below.
Also in higher dimensions the case of positive Yamabe invariant is notoriously difficult. In dimension n ≥ 5 one does not know any n-dimensional manifold M for which one can prove 0 < σ(M ) < σ(S n ). In dimensions n ≤ 4 there are some examples for which exact calculations can be carried out, even in the positive case. The values for CP 2 and some related spaces were calculated by LeBrun [20] using Seiberg-Witten theory. The calculation then was simplified considerably by Gursky and LeBrun [14]. This proof no longer uses Seiberg-Witten theory, but only the index theorem by Atiyah and Singer. See also [14,19,21] for related results.
Recently, surgery techniques known from the work of Gromov and Lawson could be refined to obtain explicit positive lower bounds for the Yamabe invariant. Such bounds are easily obtained for special manifolds, e. g. for manifolds with Einstein metrics or connected sum of such manifolds. Namely, a theorem by Obata [24] states that the Einstein-Hilbert functional of an Einstein metric g equals µ(M, [g]), thus providing a lower bound for σ(M ). For instance, if M is S n , T n , RP n or CP n , the canonical Einstein metrics provide lower bounds for σ(M ). However, obtaining a lower bound for σ(M ) is difficult in general if M carries a metric of positive scalar curvature but no Einstein metric. Using surgery theory, Petean and Yun have proven that σ(M ) ≥ 0 for all simply-connected manifolds of dimension at least 5, see [26], [27]. Stronger results can be obtained with the surgery formula developed in [2]. For example, it now can be shown, see [4] and [3,5], that simplyconnected manifolds of dimension 5 resp. 6 satisfy σ(M ) ≥ 45.1 resp. σ(M ) ≥ 49.9.

Overview over the G-equivariant Yamabe invariant
In this paper, we study the G-equivariant setting by taking the supremum and the infimum only among G-invariant metrics and conformal classes where G is a compact Lie group acting on M , see Section 3.1 for details. The associated invariants are called the G-equivariant Yamabe constant or simply the G-Yamabe constant µ(M, [g] G ), and similarly the G-(equivariant) Yamabe invariant σ G (M ). To our knowledge the first reference for the G-equivariant Yamabe constant µ(M, [g] G ) is Bérard Bergery [11]. In particular, he formulated a G-equivariant version of the Yamabe conjecture, which was the main subject of an article by Hebey and Vaugon [16] and by the second author [22,23]. In general neither σ G (M ) ≤ σ(M ) nor σ G (M ) ≥ σ(M ), see Example 3.
One motivation for the present article is to shed new light on Schoen's conjecture which is equivalent to saying σ Γ (S n ) = σ(S n ). A proof of Schoen's conjecture (or even partial results) would be very helpful, as it would provide interesting conclusions about the Yamabe invariant of non-simply connected manifolds. For example, if we were able to obtain an upper bound on σ Γ (S n ) which is uniform in Γ, then the Yamabe invariant would define interesting subgroups of the spin bordism and oriented bordism groups, see [2].
The simplest case of Schoen's conjecture is when Z k ⊂ S 1 ⊂ C acts by complex multiplication on S 3 ⊂ C 2 , the so-called Hopf action. As it seems currently out of reach to show σ Z k (S n ) = σ(S n ) for k > 2, we study the limit k → ∞ instead, and this leads two the following two questions: (1) Is σ S 1 (S 3 ) = σ(S 3 ) true for the Hopf action? (2) Assume that a sequence (H i ) of subgroups of G "converges" to G. Can we conclude that σ Hi (M ) converges to σ G (M )?
The answer to the first question is answered affirmatively by our main theorem. More generally, we give an upper bound for the S 1 -Yamabe invariant of any 3dimensional closed oriented manifold M , endowed with an S 1 -action. This upper bound depends only on the following topological invariants: the first Chern class of the associated line bundle and the Euler-Poincaré characteristic of the quotient space (see Theorem 9 for the precise statement).
Our strategy is to use the quotient space M/S 1 . We distinguish the following three cases, since the isotropy group of any point is either {id}, Z k or S 1 . If the S 1 -action has at least one fixed point, a result of Hebey and Vaugon [16] implies that σ S 1 (M ) ≤ σ(S 3 ). If the S 1 -action is free, then M/S 1 is a smooth surface. In order to find an upper bound in this case, we mainly use O'Neill's formula relating the curvatures of the total space and the base space of a Riemannian submersion and the Gauß-Bonnet theorem. In the last case, when the S 1 -action is neither free nor has fixed points (i.e. there exists at least one point with non-trivial finite isotropy group), the quotient space M/S 1 is a closed 2-dimensional orbifold. We proceed as in the free action case, since the Gauß-Bonnet theorem still holds on orbifolds (see [28]). In the two latter cases, we find a topological upper bound of σ S 1 (M ), which depends only on the Euler-Poincaré characteristic of M/S 1 and the first Chern number of the associated line bundle over M/S 1 .
The last part of the article partially answers the second question. More precisely the statement of Corollary 13 is Unfortunately, the corresponding ≤-inequality which would allow the interesting application to Schoen's conjecture still fails due to lack of curvature control.

Preliminaries, definitions and some known results
3.1. Definition of the G-equivariant Yamabe invariant. In this section we assume that a compact Lie group G acts on the compact manifold M . All actions are supposed to be smooth.
We recall that the Einstein-Hilbert functional of M is given by We denote by [g] G the set of G-invariant metrics in the conformal class ofg and by C G (M ) the set of all conformal classes containing at least one G-invariant metric.
Definition 1 (G-Yamabe invariant). We define the G-equivariant Yamabe constant (or shorter: the G-Yamabe constant) by and the G-equivariant Yamabe invariant of M (or shorter: the G-Yamabe invariant) by  Proposition 4 (Hebey-Vaugon). Let M be an n-dimensional compact connected oriented manifold endowed with an action of a compact Lie group G, admitting at least one orbit of finite cardinality. Then the following inequality holds: Other results in the literarture can be rephrased as follows.
Proposition 5 (Bérard Bergery, [11]). If G is a compact Lie group whose connected component of the identity is non-ablian and which acts effectively on a closed manifold M with cohomogeneity 2. Then σ G (M ) > 0. More recent progress about the question whether σ G (M ) > 0 can be found in [15] and [32].
3.3. Scalar curvature of S 1 -bundles. Let M n be a compact oriented and connected manifold, which is an S 1 -bundle over N , let π : M → N be the projection, letg be an S 1 -invariant metric on M and g its projection under π on N . Let K denote the tangent vector field induced by the S 1 -action and let ℓ be its length (with respect tog) and e 0 := K ℓ . We define the (2, 1)-tensor fields A and T on M as in [12, 9.C.], i.e. for all vector fields U, V on M : where H and V denote the horizontal, resp. vertical part of a vector field. The tensor A measures the non-integrability of the horizontal distribution, whereas T is essentially the second fundamental form of the S 1 -orbits. Cf. [12, 9.37], the following formula relating the scalar curvatures of (M,g) and (N, g) holds: whereδ is the codifferential in the horizontal direction. For any vector fields X, Y on N with horizontal lifts X, Y , the vertical part of [ X, Y ] equals Ω(X, Y )K := 2A X Y . We compute: where ∆ g is the Laplacian of the base (N, g).

An analytical ingredient.
We recall that the following classical result still holds on orbifolds: Then there exists a solution u ∈ C k+2 (Σ) of the equation ∆ g u = f , which is unique up to an additive constant.
The proof of Lemma 7 is analogously to the classical case.

The S 1 -Yamabe invariant
In this section we always have G = S 1 , and we use the notation N = M/S 1 similar to Section 3.3. Here N may have singular points, i.e. orbifold points or boundary points. 4.1. Yamabe functional on S 1 -bundles. If the action of S 1 is free, then by (3), we obtain from (1): since the length of any fibre is 2πℓ. The Yamabe functional of (M,g) is the restriction of the Einstein-Hilbert functional to the conformal class ofg. It can be equivalently written as follows: where Scalg is given by (3). Note that every finite quotient of S 3 by a freely acting subgroup of SO(4) admits a non-trivial S 1 -action [25, Sec. 6, Theorem 5].

4.3.
Oriented 3-manifolds. From now on, we assume that M is a 3-dimensional compact oriented connected manifold endowed with an S 1 -action. If this S 1 -action has at least one fixed point, Proposition 4 implies that the Yamabe invariant of S 3 is an upper bound for the S 1 -Yamabe invariant: σ S 1 (M ) ≤ σ(S 3 ).
We want to determine an upper bound for the S 1 -Yamabe invariant in the complementary case, i.e. we consider S 1 -actions without fixed points. This implies that M is an S 1 -principal (orbi)bundle over Σ := M/S 1 , which is a 2-dimensional orbifold (a smooth surface, if the action is free). As usually, we use the correspondence between S 1 -principal bundles and complex line bundles defined by Σ → L := Σ × S 1 C.
We are now ready to state our main result: Theorem 9. Let M be a 3-dimensional compact connected oriented manifold endowed with an S 1 -action without fixed points. With the above notation, the following assertions hold: i) If χ(Σ) > 0 and c 1 (L, Σ) = 0, then Proof of Theorem 9. Let [g] S 1 ∈ Conf S 1 (M ) be the class of S 1 -invariant metrics conformal tog on M . Without loss of generality, we assume that the length of the vector field K generating the S 1 -action ℓ := |K|g is constant (otherwise we take a different representant of the class [g] S 1 ). Let g be the projection of the metricg on Σ, so that (M,g) → (Σ, g) is a Riemannian submersion. Since ℓ is constant, the O'Neill formula (3) yields that Scalg = Scal g − ℓ 2 4 |Ω| 2 g . Using the Gauß-Bonnet theorem, we compute the Yamabe functional as follows: If we have χ(Σ) ≥ 0 and Ω 2 > 0, then the maximal value of this expression as a function in ℓ is attained for ℓ = 4πχ(Σ) Ω −1 2 and its maximal value equals 2 . We now consider cases i) to iii) in the theorem. i) Note that in this case c 1 (L, Σ) = 0 implies Ω 2 > 0. By the Cauchy-Schwarz inequality, it further follows that On the other hand, we claim that Ω 1 ≥ 2 √ 2π|c 1 (L, Σ)|, since where the volume form dv g has length √ 2, by convention. Using σ(S 3 ) = 3 ·2 5/3 ·π 4/3 it follows that J(g) ≤ σ(S 3 )χ(Σ) Scal g has zero average over Σ. By Lemma 7, there exists a solution u of the equation ∆ g u = f . Therefore the scalar curvature of g u := e 2u g is given by Hence, the scalar curvature of g u is positive. Using the identity (3) and choosing the length of the S 1 -fibre constant and sufficiently small, we construct an S 1 -invariant metricg u (which is not necessarily conformal tog) with positive scalar curvature. Therefore, the Yamabe constant µ(M, [g u ] S 1 ) is positive, so σ S 1 (M ) > 0. ii) If c 1 (L, Σ) = 0, then there exists an S 1 -equivariant finite covering S 1 × Σ of M of degree d, where Σ is a smooth compact surface finitely covering Σ (for more details, see e.g. [30, Lemma 3.7]). Since χ(Σ) > 0, we see that Σ is diffeomorphic to S 2 . As in the previous case, we know that a metric of positive Gauß curvature exists on Σ. The product metricg ℓ of its lift to Σ with a rescaled standard metric on S 1 of length 2πℓ is invariant under the deck transformation group of S 1 × Σ → M . As this deck transformation group commutes with the S 1 -action,g ℓ descends to an S 1 -invariant metric g ℓ on M . From (5), we get . Then µ(M, [g ℓ ] S 1 ) converges to ∞ for ℓ → ∞, which implies the statement. iii) Assume that the Euler-Poincaré characteristic of Σ is nonpositive. By (6), we have for any S 1 -invariant Riemannian metricĝ on M , wherel := |K|ĝ. This yields σ S 1 (M ) ≤ 0. Moreover, if we fix a Riemannian metricĝ Σ on Σ, we define (ĝ j ) to be a sequence of metrics on M with constant functionsl j := |K|ĝ j ≤ 1 converging to 0 and π * ĝ Σ =ĝ j . From (5) and using the Hölder inequality, we obtain Hence, when j goes to +∞, it follows that σ S 1 (M ) ≥ 0. We conclude that σ S 1 (M ) = 0.
4.4. The case of S 3 . We now consider the special case of S 1 -actions on S 3 ⊂ C 2 .
For m 1 , m 2 ∈ N assumed to be relatively prime as long as m 1 m 2 = 0, we define With this notation, the Hopf action of S 1 on S 3 corresponds to φ 1,1 . These are the only possible smooth S 1 -actions on S 3 up to diffeomorphisms (see e.g. [25]). Note that such an action has fixed points if and only if m 1 m 2 = 0.
Theorem 10. For the Hopf action of S 1 on S 3 it holds: Moreover, the S 1 -equivariant Yamabe invariant of any S 1 -action φ m1,m2 on S 3 satisfies the following: Proof. Let us first remark that, since the standard metric g st on S 3 is S 1 -invariant for any S 1 -action φ m1,m2 , it follows that µ(S 3 , Hence, we obtain the inequality: σ S 1 (S 3 ) ≥ σ(S 3 ). i) If m 1 m 2 = 0, then the S 1 -action has fixed points and by Proposition 4 we also obtain the reverse inequality: σ S 1 (S 3 ) ≤ σ(S 3 ). ii) If m 1 m 2 = 0, then the quotient orbifold is the so-called 1-dimensional weighted projective space denoted by CP 1 (m 1 , m 2 ). In order to use the upper bound provided by Theorem 9, we need to compute χ(CP 1 (m 1 , m 2 )) and c 1 (L, CP 1 (m 1 , m 2 )).

Convergence result
Definition 11. Let G be a Lie group, and let (H i ) i∈N be a sequence of subgroups. We say that h ∈ G is an accumulation point for (H i ) i∈N if there is a sequence (h i ) i∈N with h i ∈ H i and h i → h. The set of accumulation points is a closed subgroup of G. We say that (H i ) i∈N is accumulating, if every element of G is an accumulation point. . Obviously µ i ≤ µ. After passing to a subsequence we can assume that µ i converges to a numberμ ≤ µ and it remains to show that µ < µ leads to a contradiction. For an orbit O we will use the convention that #O takes values in N ∪ {∞}, i.e. we do not distinguish between infinite cardinalities. We claim that lim i→∞ #(H i · p) = #(G · p), for any p ∈ M . The inequality To prove the claim in the case #(G · p) < ∞, we choose pairwise disjoint neighborhoods of all the G-orbit points of p and for i sufficiently large, we find in each such neighborhood an element of the H i -orbit of p, showing that #(H i · p) ≥ #(G · p). If #(G · p) = ∞, then we apply the previous argument to a finite subset of the G-orbit of p and then let its cardinality converge to ∞. This shows that lim i→∞ #(H i · p) = ∞.
Without loss of generality, we assume that µ i ≤μ := (µ +μ)/2 < µ. Let k be the cardinality of the smallest G-orbit, again sloppily written as ∞ in the case that k is infinite. Then by Proposition 4, we have µ ≤ σ(S n )k 2/n . This implies µ i ≤μ < σ(S n )k 2/n . Hence, by the above claim, we obtain the following inequality µ i ≤μ < σ(S n )(min p∈M #(H i ·p)) 2/n , for i ≥ i 0 , where i 0 is sufficiently large. By Hebey and Vaugon [16], it follows that, for i ≥ i 0 , there exists a sequence (u 4 n−2 i g) i∈N of H i -invariant metrics, which minimizes the functional J among all H i -invariant metrics in [g]. Furthermore u i is a positive smooth H i -invariant solution of the Yamabe equation, and we may assume u i 2n n−2 = 1. The sequence (u i ) i∈N is uniformly bounded in H 1 (M ). Hence there exists a nonnegative function u ∞ ∈ H 1 (M ), such that (u i ) i∈N converges strongly in L q (M ), for 1 ≤ q < 2n n−2 , and weakly in H 1 (M ) to u ∞ . We now claim, that u i is bounded in the L ∞ -norm. Suppose that it is not bounded. Then we find a sequence of x i ∈ M such that u i (x i ) → ∞, and after taking a subsequence x i converges to a pointx. For any point gx in its orbit, there is a sequence of If the orbit G·x contains at leastk points, then we can do classical blowup-analysis ink points, which would yieldμ ≥ σ(S n )k 2/n (see for example [10, Chapter 6.5.]). This impliesμ ≥ σ(S n )k 2/n which contradictsμ < σ(S n )k 2/n . We obtain the claim, i.e. the boundedness of u i in L ∞ . By a standard bootstrap argument this yields the boundedness of u i in C 2,α for 0 < α < 1, and thus u i converges to u ∞ in C 2 . It follows that u ∞ is a smooth, positive G-invariant solution of the Yamabe equation, with u ∞ 2n n−2 = 1 and J(u Appendix A. A.1. Computation of c 1 (L, CP 1 (m 1 , m 2 )). We consider the action of S 1 on S 3 ⊂ C 2 given by where m 1 and m 2 are two positive relatively prime integers. Let π : S 3 → S 3 /S 1 denote the projection, where the quotient S 3 /S 1 =: CP 1 (m 1 , m 2 ) is the one dimensional weighted projective space. We consider the round metric of S 3 induced by the standard metric on R 4 ≃ C 2 : (z 1 , z 2 ), (w 1 , w 2 ) = Re(z 1w1 + z 2w2 ) . The vector field induced by the S 1 -action is given by: The vector field K vanishes nowhere, since |K p | 2 = m 2 , the orthogonal complement of K p in T p S 3 (w.r.t. the round metric) is spanned by the horizontal vector fields X 1 (p) := i(m 2 |z 2 | 2 z 1 , −m 1 |z 1 | 2 z 2 ), X 2 (p) := (|z 2 | 2 z 1 , −|z 1 | 2 z 2 ), which are also S 1 -invariant. Hence they project to the vector fields X 1 , resp. X 2 on CP 1 (m 1 , m 2 ).