3-Manifolds with Positive Flat Conformal Structure

In this paper, we consider a closed 3-manifold $M$ with flat conformal structure $C$. We will prove that, if the Yamabe constant of $(M, C)$ is positive, then $(M, C)$ is Kleinian.


Introduction and Main Theorem
In 1988, Schoen and Yau [19] gave a final resolution for the Yamabe Problem (cf. [3,15,18]). In [19,Proposition 3.3], they also proved that any closed n-manifold with flat conformal structure of positive Yamabe constant is Kleinian, provided that n ≥ 4. Moreover, under the assumption that an extended Positive Mass Theorem holds (but a proof has not yet appeared), they showed that the above assertion still holds even when n = 3 (see [19,Proposition 4.4 ′ ] and the paragraph just before it). On the other hand, there are enormous examples of closed 3-manifolds with flat conformal structures which are not Kleinian (see [8,Remark 7.4]).
The purpose of this brief note is to prove the above assertion for the remaining case n = 3. This assertion can be obtained by an argument in the proof of [1, the second assertion of Theorem 1.4], which is a combination of a result [19,Proposition 4.2], a positive mass theorem [1, the first assertion of Theorem 1.4] (different form the one Schoen and Yau mentioned in [19]) and a classification of 3-manifolds with positive scalar curvature [7,10,11]. Here, we will explicitly give a proof of it.
The remaining sections are organized as follows. Section 2 contains some necessary definitions and preliminary geometric results. Section 3 is devoted to the proof of Theorem 1.1. Acknowledgements. The second author would like to thank Hiroyasu Izeki and Gilles Carron for helpful discussions and for useful comments respectively.

Preliminaries
Let M be a closed 3-manifold, that is, a compact 3-manifold without boundary. To simplify the presentation and the argument, we always assume that dim M = 3 Date: March, 2011. * supported in part by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 21540097. throughout this paper. For each conformal class C on M , the Yamabe constant Y (M, C) of (M, C) is defined by where R g , µ g and Vol g (M ) denote respectively the scalar curvature, the volume element of g and the volume of (M, g). It is a finite-valued conformal invariant of C. The Yamabe constant Y (M, C) is positive if and only if there exists a positive scalar curvature metric g ∈ C (cf. [3]). A remarkable theorem [22,20,2,17,19] of Yamabe, Trudinger, Aubin and Schoen asserts that each conformal class C contains a minimizerǧ of E| C , called a Yamabe metric (or a solution of the Yamabe Problem), which is of constant scalar curvature Let M ∞ is an infinite covering of M . We shall call that the fundamental group π 1 (M ) of M has a descending chain of finite index subgroups tending to π 1 (M ∞ ) if it satisfies the following: There exists a family of subgroups Assume that Y (M, C) > 0. Take a positive scalar curvature metric g ∈ C and any point p ∈ M . Then, there exists the normalized Green's function G p for L g with a pole at p, that is, Here, L g := −8∆ g + R g , c 0 > 0 and δ p stand respectively for the conformal Laplacian, a specific universal positive constant and the Dirac δ-function at p. Assume also that the covering P ∞ : M ∞ → M is normal. Let g ∞ denote the lift of g to M ∞ , and p ∞ a point in M ∞ with P ∞ (p ∞ ) = p. Then, there exists uniquely also a normalized minimal positive Green's function G ∞ on M ∞ for L g∞ := −8∆ g∞ +R g∞ with pole at p ∞ (cf. [19]), which satisfies the following Here, G stands for the group of deck transformations for the normal covering M ∞ → M . Set Then, g ∞,AF defines a scalar-flat, asymptotically flat metric on M * ∞ (cf. [15]). Note that this asymptotically flat 3-manifold (M * ∞ , g ∞,AF ) has infinitely many singularities created by the ends of M * ∞ . However, the mass m ADM (g ∞,AF ) of (M * ∞ , g ∞,AF ) can be defined in the usual way (cf. [4]). Note also that the positive mass theorem for asymptotically flat 3-manifolds with singularities does not always hold (see [1, Remark 1.5-(2)] for instance).
With these understanding, the following positive mass theorem holds as a special case of [1, the first assertion of Theorem 1.4]: be a normal infinite Riemannian covering of (M, g) such that π 1 (M ) has a descending chain of finite index subgroups tending to π 1 (M ∞ ), where g ∈ C is a positive scalar curvature metric and g ∞ is its lift to M ∞ . For any point p ∞ ∈ M ∞ , let G ∞ denote the normalized minimal positive Green's function on M * ∞ with pole at p ∞ . Then, the asymptotically flat 3-manifold (M * ∞ , g ∞,AF ) has nonnegative mass m ADM (g ∞,AF ) ≥ 0.
Remark 2.2. Assume that M = #ℓ(S 1 × S 2 ) for ℓ ≥ 2 and M ∞ is its universal covering. Note that M ∞ is spin. For each small σ > 0, consider the complete metric g σ,AF : . Then, only one end of (M * ∞ , g σ,AF ) is asymptotically flat and the other infinitely many ends are merely complete. For the authors, it is not clear whether Witten's approach [21] (cf. [16]) to Positive Mass Theorem is still valid for (M * ∞ , g σ,AF ). Hence, we will use here Proposition 2.1 for the proof.
A conformal 3-manifold (M, C) is is said to be locally conformally flat if, for any point p ∈ M , there exists a metric g ∈ C such that g is flat on some neighborhood of p. A conformal class C on M is called a flat conformal structure if (M, C) is locally conformally flat. In [14], Kuiper proved that, for a simply connected locally conformally flat 3-manifold (X, C ′ ), there is a conformal immersion into (S 3 , C 0 ) called developing map, which is unique up to composition with a Möbius transformation of (S 3 , C 0 ). Therefore, the universal covering of a locally conformally flat manifold (M, C) admits a developing map. Here, (S 3 , C 0 ) denotes the 3-sphere

Proof of Main Theorem
Proof of Theorem 1.1. Consider the universal covering M of M and denote the lift of the flat conformal structure C by C. If |π 1 (M )| < ∞, then ( M , C) is conformal to (S 3 , C 0 ) by Kuiper's Theorem [14]. Hence, (M, C) is Kleinian. From now on, we assume that |π 1 (M )| = ∞, that is, the degree of the covering map P : M → M is infinite.
Take a unit-volume Yamabe metric g ∈ C, and consider its lift g ∈ C to M . Note that R g = R g = Y (M, C) > 0. Take any base points p ∈ M, p ∈ M satisfying P ( p) = p, and fix them. Then, let G denote the normalized minimal positive Green function on M for L g with pole at p, and the mass m ADM ( g AF ) of the asymptotically flat 3-manifold ( M − { p}, g AF := G 4 · g).
Suppose that m ADM ( g AF ) ≥ 0. Recall that we can choose the base point p ∈ M arbitrarily. It then follows from Proposition 2.3 that the developing map of ( M , C) is injective, and hence (M, C) is Kleinian. In this case, especially m ADM ( g AF ) = 0. Therefore, it is enough to show m ADM ( g AF ) ≥ 0.
Recall that M is the infinite universal covering of M . Then, there exists (uniquely) an infinite universal covering M → M ′ . Moreover, since π 1 (M ′ ) is a finitely generated free group, it has a descending chain of finite index subgroups tending to π 1 ( M ) = {e}. Let g ′ be the lifting of g to M ′ . Applying Proposition 2.1 to the normal infinite Riemannian covering ( M , g) → (M ′ , g ′ ), we have that m ADM ( g AF ) ≥ 0. This completes the proof of Theorem 1.1.
Remark 3.1. Even if we replace the positivity Y (M, C) > 0 in Theorem 1.1 by the nonnegativity Y (M, C) ≥ 0, it seems that the same conclusion still holds. More precisely, we propose the following (cf. [5,13]).
Conjecture. Let M be a closed 3-manifold with flat conformal structure C. If its Yamabe constant is zero, then either of the following (1) or (2) holds: (1) There exists a flat metric g ∈ C.
In the case (1), the universal covering ( M , C) of (M, C) is conformal to (S 3 − {p N }, C 0 ) where p N := (1, 0, 0, 0) ∈ S 3 , and hence (M, C) is Kleinian. In the case (2), Theorem 1.1 implies that (M, [g 1 ]) is Kleinian. The argument in Proof of Theorem 1.1 also implies that there exists a torsion free subgroup Γ of finite index in π 1 (M ) such that Γ is either a trivial group or a non-trivial finitely generated free group. Then, the virtual cohomological dimension vcd π 1 (M ) of π 1 (M ) is either 0 or 1 (see [6]). Therefore, (M, [g 1 ]) is a closed Kleinian 3-manifold with vcd π 1 (M ) < 3. The quasiconformal stability of Kleinian groups [12,Theorem 2] implies that any flat conformal structure on M which is a smooth deformation of [g 1 ] is also Kleinian, particularly C is too.