ON POSITIVE SCALAR CURVATURE AND MODULI OF CURVES

Transcription

1 ON POSIIVE SCALAR CURVAURE AND MODULI OF CURVES KEFENG LIU AND YUNHUI WU Abstract. In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus g with g 2 does not admit any Riemannian metric ds 2 of nonnegative scalar curvature such that ds 2 ds 2 where ds 2 is the eichmüller metric. Our second result is the proof that any cover M of the moduli space M g of a closed Riemann surface S g does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasiisometry class of the eichmüller metric, which implies a conjecture of Farb-Weinberger in [Far06]. 1. Introduction Many aspects of positive scalar curvatures on Riemannian manifolds have been well understood since the fundamental works of Schoen-Yau [SY79b, SY79a] and Gromov-Lawson [GL80, GL83]. Important generalizations have been developed by Roe [Roe93], Yu [Yu98] and many others. he main object of this paper is to study obstructions to the existence of positive scalar curvature metric on the moduli spaces of Riemann surfaces. Let S g be a closed Riemann surface of genus g with g 2, Mod(S g ) be the mapping class group and eich(s g ) be the eichmüller space of S g. opologically eich(s g ) is a manifold of real dimension 6g 6, which carries various Mod(S g )-invariant metrics which descend to metrics on the moduli space M g of S g with respective properties. For examples, the famous Weil- Petersson metric and eichmüller metric. he eichmüller metric ds 2 is not Riemannian but is a complete Finsler metric. It was shown in [Mas75] that ds 2 is not nonpositively curved in the metric sense by showing that there exists two different geodesic rays starting at the same point such that they have bounded Hausdorff distance. Furthermore, Masur and Wolf in [MW95] showed that (eich(s g ), ds 2 ) is not Gromov-hyperbolic. he Weil-Petersson metric ds 2 W P is Kähler [Ahl61], incomplete [Chu76, Wol75], geodesically convex [Wol87] and has negative sectional curvature [Wol86, ro86]. Brock and Farb in [BF06] showed that the space (eich(s g ), ds 2 W P ) is also not Gromov-hyperbolic Mathematics Subject Classification. 30F60, 32G15, 53C21. Key words and phrases. Moduli space, Scalar curvature, eichmüller metric, Riemannian metric. 1

2 2 KEFENG LIU AND YUNHUI WU here are also some other important metrics on M g. For examples, the asymptotic Poincaré metric, the induced Bergman metric, the Kähler- Einstein metric, the McMullen metric, the Ricci metric, and the perturbed Ricci metric are all complete and Kähler metrics. he Kobayashi metric and the Caratheódory metric are complete and Finsler metrics. In [LSY04, LSY05, McM00], the authors showed that all these metrics are bi- Lipschitz (or equivalent) to the eichmüller metric. And the Weil-Petersson metric plays an important role in their proofs. he perturbed Ricci metric [LSY04, LSY05] has pinched negative Ricci curvature. In particular it also has negative scalar curvature. he McMullen metric [McM00] has negative scalar curvature at certain points since the metric, restricted on certain thick part of the moduli space, is the Weil- Petersson metric. However, Farb and Weinberger in [FW] showed that any finite cover M of the moduli space M g (g 2) admits a complete finitevolume Riemannian metric of (uniformly bounded) positive scalar curvature, which is analogous to Block-Weinberger s result in [BW99] on certain locally symmetric arithmetic manifolds. he second author in [Wu15] applied the negativity of the Ricci curvature of the perturbed Ricci metric [LSY04, LSY05] to show that any finite cover of the moduli space M g does not admit any complete finite-volume Hermitian metric of nonnegative scalar curvature. Moreover, he also showed that the total scalar curvature of any almost Hermitian metric, which is bi-lipschitz (or equivalent) to the eichmüller metric, is negative provided the scalar curvature is bounded from below. However, the method in [Wu15] highly depends on the canonical complex structure on M g, which fails in the setting of Riemannian metrics. Our aim in this article is to study the Riemannian case. Before stating the results we need to fix some notations. Let ds 2 be the eichmüller metric on the eichmüller space on which the mapping class group acts properly by isometries. hus, ds 2 descends to a metric on the moduli space M g, which is still denoted by ds 2. We let M be any cover of M g. he metric ds 2 can be naturally lifted to a metric on M, still denoted by ds 2 and called the eichmüller metric on M. Let ds2 be a Riemannian metric on M. We say ds 2 ds 2 if there exists a constant k > 0 such that ds 2 k ds 2 on the tangent bundle. hat is, for any p M and v pm we have v ds 2 k v ds 2. One of Gromov-Lawson s theorems in [GL83] (one can also see heorem 1.1 in [Roe93]) says that heorem 1.1 (Gromov-Lawson). Let (X, ds 2 1 ) be a complete Riemannian manifold of nonpositive sectional curvature. hen for any Riemannian metric ds 2 2 on X with ds2 2 ds2 1, the Riemannian manifold (X, ds2 2 ) cannot have positive scalar curvature on X.

3 SCALAR CURVAURE 3 One immediate application of heorem 1.1 is that the torus n (n 2) can not carry a complete Riemannian metric of positive scalar curvature, which answers a question of Geroch. For low dimensions n 7, this was first settled in a series of papers by R. Schoen and S.. Yau in [SY79b] and [SY79a]. Our first result in this paper is heorem 1.2. Let S g be a closed Riemann surface of genus g with g 2 and M be a finite cover of the moduli space M g of S g. hen for any Riemannian metric ds 2 on M with ds 2 ds 2 we have inf Sca(p) < 0. p (M,ds 2 ) As stated above the eichmüller metric is not nonpositively curved. So the argument in [GL83] can not lead to heorem 1.2. We are going to use some recent developments in [McM00, LSY04, LSY05, BBF14] on the geometry of eichmüller space as bridges to prove heorem 1.2. One can see more details in Section 6. Let ds 2 M be the McMullen metric and ds2 LSY be the perturbed Ricci metric. We let ds 2 a be an arbitrary Riemannian metric on M. heorem 1.2 applies to the metrics ds 2 = ds 2 M + ds2 a and ds 2 = ds 2 LSY + ds2 a, which are not covered in [Wu15] if ds 2 a is only Riemannian and not Hermitian. We remark here that there is no finite-volume condition for heorem 1.2. As stated above Farb and Weinberger in [FW] proved the existence of complete Riemannian metrics of uniformly positive scalar curvatures on the moduli space. Actually they also showed that these metrics are not quasiisometric to the eichmüller metric. Motivated by Chang s result in [Cha01] on certain locally symmetric spaces, the following conjecture is posed in [Far06] (see Conjecture 4.6 in [Far06]), Conjecture 1.3 (Farb-Weinberger). Let S g be a closed surface of genus g with g 2. hen any finite cover M of the moduli space M g of S g does not admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the quasi-isometry class of the eichmüller metric. Definition 1.4. Let M be a cover of the moduli space M g and ds 2 be a Riemannian metric on M. Here the cover may be an infinite cover. We call that ds 2 on M is quasi-isometric to the eichmüller metric ds 2 if there exists two positive constants L 1 and K 0 such that on the universal cover (eich(s g ), ds 2 ) ((eich(s g ), ds 2 )) of (M, ds2 ) ((M, ds 2 )) respectively, the identity map satisfies L 1 dist ds 2 (p, q) K dist ds 2(p, q) L dist ds 2 (p, q)+k, p, q eich(s g ). Our second result is the following one, which in particular gives a proof of Conjecture 1.3 of Farb-Weinberger.

4 4 KEFENG LIU AND YUNHUI WU heorem 1.5. Let S g be a closed surface of genus g with g 2. hen any cover M of the moduli space M g of S g does not admit a complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the eichmüller metric. he theorem above clearly implies Conjecture 1.3. We remark here that there are no conditions on finite cover and finite volume in heorem 1.5, compared to the Farb-Weinberger conjecture. We remark here that Farb and Weinberger have a different approach to Conjecture 1.3 by using methods from Chang s thesis [Cha01] together with a theorem of Farb-Masur in [FM10] on the asymptotic cone of the moduli space. We thank Prof. Farb for sharing their information. For the bi-lipschitz case instead of quasi-isometry, Conjecture 1.3 is also obtained recently by E. Leuzinger in [Leu10]. See the abstract and heorem C in [Leu10]. Both heorem 1.2 and 1.5 in this paper cover the bi-lipschitz case of Conjecture 1.3. Remark 1.6. (1). Following totally the same arguments in this article, one can deduce that both heorem 1.2 and 1.5 are still true for the eichmüller space of noncompact surface S g,n of genus g with n punctures if 3g + n 5. Note that heorem 5.1 and 5.2 require that the dimension of the space is greater than or equal to 3. (2). For the cases (g, n) = (1, 1) or (0, 4), it is not hard to see that heorem 1.2 and 1.5 still hold without the assumptions on the eichmüller metric, since the scalar curvature is exact sectional curvature for these cases. More precisely, heorem 1.2 directly follows from the fact that the mapping class group contains free subgroups of rank 2. And heorem 1.5 follows from the classical Bonnet-Myers heorem which says that a complete Riemannian manifold of uniformly positive sectional curvature is compact, while the eichmüller space is a disk which is open. (3). he existence theorem of Farb-Weinberger in [FW], which one can also see heorem 4.5 in [Far06], tells that heorem 1.2 does not hold anymore without the assumptions on the eichmüller metric if 3g + n 6. It is interesting to know whether heorem 1.2 is still true without the assumption on the eichmüller metric when 3g + n = Plan of the paper. In Section 2 we give some necessary preliminaries and notations for surface theory. In Section 3 we review some recent developments on the geometry of eichmüller space which will be served as bridges to prove heorem 1.2 and 1.5. In Sections 4 and 5 we will show that any complete Riemannian metric on the moduli space of surfaces with nonnegative scalar curvature can be deformed an equivalent Riemannian metric of positive scalar curvature. heorem 1.2 will be proved in Section 6. And in Section 7 we will establish heorem 1.5. A related open problem will be discussed in Section 8.

5 SCALAR CURVAURE 5 Acknowledgement. he authors would like to thank E. Leuzinger, G. Yu and S.. Yau for their interests. hey are also grateful to B. Farb and S. Weinberger for the conversation on heorem 1.5. he first author is supported by an NSF grant. he second author is grateful to J. Brock and M. Wolf for their consistent encouragement and help. He also would like to acknowledge support from U.S. National Science Foundation grants DMS , , RNMS: Geometric structures And Representation varieties (the GEAR Network). 2. Notations and Preliminaries 2.1. Surfaces. Let S g be a closed Riemann surface of genus g with g 2, and M 1 denote the space of Riemannian metrics on S g with constant curvature 1, and X = (S g, σ dz 2 ) be an element in M 1. he group Diff 0 of diffeomorphisms of S g isotopic to the identity, acts on M 1 by pull-backs. he eichmüller space eich(s g ) of S g is defined by the quotient space eich(s g ) = M 1 / Diff 0. Let Diff + be the group of orientation-preserving diffeomorphisms of S g. he mapping class group Mod(S g ) is defined as Mod(S g ) = Diff + / Diff 0. he moduli space M g of S g is defined by the quotient space M g = eich(s g ) /Mod(S g ). he eichmüller space has a natural complex structure, and its holomorphic cotangent space X eich(s g) is identified with the quadratic differentials QD(X) = {φ(z)dz 2 } while its holomorphic tangent space is identified with the harmonic Beltrami differentials HBD(X) = { φ(z) dz σ(z) dz } Mapping class group. he mapping class group Mod(S g ) is a finitely generated discrete group which acts properly on the eichmüller space. One special set of generators of Mod(S g ) is the Dehn-twists along simple closed curves, which play an important role in studying Mod(S g ). Let α be a nontrivial simple closed curve on S g and τ α be the Dehn-twist along α. he following lemma will be applied later. Lemma 2.1. Let α, be two simple closed curves on M such that the geometric intersection points i(α, ) 2. hen, for any n, m Z +, the group < τα n, τ m > is a free group of rank 2. Proof. One can check chapter 3 in [FM12] for details.

6 6 KEFENG LIU AND YUNHUI WU 2.3. he eichmüller metric and Weil-Petersson metric. Recall that the eichmüller metric ds 2 on eich(s g) is defined as φ(z) σ(z) φ(z) dz ds 2 := sup ψ(z)dz. ψdz 2 QD(X), X σ(z) 2i X ψ =1 Re he induced path metric of the above metric, denoted by dist, on eich(s g ) can also be characterized as follows; let p 1, p 2 eich(s g ), then dist (p 1, p 2 ) = 1 2 log K where K 1 is the least number such that there is a K-quasiconformal mapping between the hyperbolic surfaces p 1 and p 2. he eichmüller metric is not Riemannian but Finsler. he following fundamental theorem on the eichmüller metric will be used later. heorem 2.2 (eichmüller). (1). he eichmüller space (eich(s g ), ds 2 ) is complete. (2). he eichmüller space (eich(s g ), ds 2 ) is uniquely geodesical, i.e., for any two points p 1, p 2 eich(s g ) there exists a unique geodesic c : [0, 1] (eich(s g ), ds 2 ) such that c(0) = p 1 and c(1) = p 2. A direct corollary is Proposition 2.3. Any geodesic ball of finite radius in (eich(s g ), ds 2 ) is contractible. Proof. Let p (eich(s g ), ds 2 ) and r > 0. Consider the geodesic ball B(p; r) := {q; dist (p, q) r} (eich(s g ), ds 2 ). For any z B(p; r), by heorem 2.2 we know that there exists a unique geodesic c z : [0, dist (p, z)] B(p; r) such that c z (0) = p and c z (dist (p, z)) = z. Here we use the arc-length parameter for c. hen we consider the following map H : B(p; r) [0, 1] B(p; r) (z, t) c z (t dist (p, z)). heorem 2.2 tells us that H is well-defined and continuous. It is clear that H(z, 0) = p and H(z, 1) = z z B(p; r). hat is, B(p, r) is contractible. For more details on eichmüller geometry, one can refer to the book [I92] and the recent survey [Mas] for more details.

7 SCALAR CURVAURE 7 he Weil-Petersson metric ds 2 W P is the Hermitian metric on g arising from the the Petersson scalar product < ϕ, ψ > ds 2 = W P X ϕ(z) ψ(z) dz dz σ(z) 2i via duality. he Weil-Petersson metric is Kähler ([Ahl61]), incomplete ([Chu76, Wol75]) and has negative sectional curvature ([Wol86, ro86]). One can refer to Wolpert s recent book [Wol10] for the progress on the study of the Weil-Petersson metric. Both the eichmüller metric and the Weil-Petersson metric are Mod(S g )- invariant. Let ds 2 1 and ds2 2 be any two Riemannian metrics on eich(s g). If there exists a constant k > 0 such that ds 2 1 k ds 2 2, then we write ds 2 1 ds 2 2. We call the two metrics ds 2 1 and ds2 2 are bi-lipschitz (or equivalent) if ds 2 1 ds 2 2 and ds 2 2 ds 2 1 which is denoted by ds 2 1 ds 2 2. It is not hard to see that the Cauchy-Schwarz inequality and the Gauss- Bonnet formula gives that ds 2 ds 2 W P. However, since the Weil-Petersson metric is incomplete and the eichmüller metric is complete, we have ds 2 W P ds 2. We close this section by introducing the following concept. Let dist ds 2 1 and dist ds 2 2 be the induced path metrics of ds 2 1 and ds2 2 on eich(s g) respectively. Let L 1 and K 0 be two constants. We call the two metrics ds 2 1 and ds 2 2 are (L,K)-quasi-isometric, if for any two points p, q eich(s g), (2.1) L 1 dist ds 2 2 (p, q) K dist ds 2 1 (p, q) L dist ds 2 2 (p, q) + K. It is clear that two metrics ds 2 1 and ds2 2 on eich(s g) are equivalent if and only if ds 2 1 and ds2 2 are (L,0)-quasi-isometric for some L 1.

8 8 KEFENG LIU AND YUNHUI WU 3. Universal properties of Riemannian metrics equivalent to ds 2 It is shown in [LSY04, LSY05, McM00] that the asymptotic Poincaré metric, the induced Bergman metric, Kähler-Einstein metric, the McMullen metric, the Ricci metric, and the perturbed Ricci metric are all Kähler and equivalent to the eichmüller metric. Actually for any metric ds 2 in the convex hull of all these metrics we have ds 2 ds 2. We are going to apply one of these metrics as bridges to prove heorems 1.2 and 1.5. We remark here that certain universal properties of the six metrics above are enough in this article. We do not need certain special property of some metric in the list above Kähler metrics on M g. In this subsection we briefly review some properties of the following three Kähler metrics M g : the McMullen metric, the Ricci metric, and the perturbed Ricci metric. hey will be applied to prove heorem 1.2 and heorem he McMullen metric. In [McM00] McMullen is the first to construct a new metric ds 2 M on M g which is Kähler-hyperbolic in the sense of Gromov and has bounded geometry in the sense of differential geometry. We call ds 2 M the McMullen metric. More precisely, let Log : R + [0, ) be a smooth function such that (1). Log(x) = log(x) if x 2; (2). Log(x) = 0 if x 1. For suitable choices of small constants ɛ, δ > 0, the Kähler form of the McMullen metric is ω M = ω W P iδ Log ɛ l γ l γ(x)<ɛ where the sum is taken over primitive short geodesics γ on X and ω W P is the Kähler form of the Weil-Petersson metric. heorem 3.1 (McMullen). On the moduli space M g, (M, ds 2 M ) satisfies (1). (M g, ds 2 M ) has bounded sectional curvatures and finite volume. (2). ds 2 M ds2. (3). here exists a constant ɛ 0 > 0 such that the injectivity radius of the universal cover satisfies that inj(eich(s g ), ds 2 M) ɛ 0 > he Ricci metric and the perturbed Ricci metric. In [ro86, Wol86] it is shown that the Weil-Petersson metric has negative sectional curvature. he negative Ricci curvature tensor defines a new metric ds 2 τ on M g, which is called the Ricci metric. rapani in [ra92] proved ds 2 τ is a complete Kähler metric.

9 SCALAR CURVAURE 9 In [LSY04] Liu-Sun-Yau perturbed the Ricci metric with the Weil-Petersson metric to give new metrics on M g which are called the perturbed Ricci metrics, denoted by ds 2 LSY. More precisely, let ω τ be the Kähler form of the Ricci metric, for any constant C > 0, the Kähler form of the perturbed Ricci metric is ω LSY = ω τ + C ω W P. Motivated by the results in [McM00], K. Liu, X. Sun and S.. Yau in [LSY04] showed that heorem 3.2 (Liu-Sun-Yau). On the moduli space M g, both (M g, ds 2 τ ) and (M g, ds 2 LSY ) satisfy (1). hey have bounded sectional curvatures and finite volumes. (2). ds 2 τ ds 2 LSY ds2 M. (3). here exists a constant ɛ 0 > 0 such that the injectivity radii of the universal covers satisfy that and inj(eich(s g ), ds 2 τ ) ɛ 0 > 0 inj(eich(s g ), ds 2 LSY ) ɛ 0 > 0. Furthermore, in one of their subsequent papers [LSY05] they also showed that the perturbed Ricci metric ds 2 LSY has negatively pinched holomorphic sectional curvatures, which is known to be the first complete metric on the moduli space with this property. And they use this property and the Schwarz-Yau lemma to prove that a list of canonical metrics on the moduli space are equivalent Asymptotic dimension. Gromov in [Gro93] introduced the notion of asymptotic dimension as a large-scale analog of the covering dimension. More precisely, a metric space X has asymptotic dimension asydim(x) n if for every R > 0 there is a cover of X by uniformly bounded sets such that every metric R-ball intersects at most n + 1 of sets in the cover. One can refer to heorem 19 in [BD08] for some other equivalent definitions of the asymptotic dimension. By using Minsky s product theorem in [Min96] for the thin part of the eichmüller space (eich(s g ), ds 2 ), recently M. Bestvina, K. Bromberg and K. Fujiwara in [BBF14] proved the following result which is crucial for this paper. heorem 3.3 (Bestvina-Bromberg-Fujiwara). Let S g be a closed surface of genus g with g 1. hen the eichmüller space, endowed with the eichmüller metric, satisfies asydim((eich(s g ), ds 2 )) <. From the definition of the asymptotic dimension it is not hard to see that the asymptotic dimension is a quai-isometric invariance. For more details, one can see the remark on page 21 of [Gro93] or Proposition 22 in [BD08].

10 10 KEFENG LIU AND YUNHUI WU heorem 3.4. Let ds 2 be a Riemannian metric on eich(s g ) such that (eich(s g ), ds 2 ) is quasi-isometric to (eich(s g ), ds 2 ) in the sense of inequality (2.1). hen, (3.1) asydim((eich(s g ), ds 2 )) <. In particular, for either the McMullen metric ds 2 M, the Ricci metric ds2 τ, or the perturbed Ricci metric d 2 LSY we have (3.2) asydim((eich(s g ), ds 2 M)) = asydim((eich(s g ), ds 2 τ )) = asydim((eich(s g ), ds 2 LSY )) <. Proof. Since the asymptotic dimension is a quasi-isometry invariant of eich(s g ), it is clear that inequality (3.1) follows from heorem 3.3. From heorem 3.1 and heorem 3.2, either the McMullen metric ds 2 M, the Ricci metric ds 2 τ, or the perturbed Ricci metric d 2 LSY is bi-lipschitz to the eichmüller metric ds 2. In particular, they are quasi-isometric to ds2. hen equation (3.2) follows. 4. No Ricci-Flat Let M be a finite cover of the moduli space M g and ds 2 be a complete Riemannian metric on M. Since M may be an orbiford, the Riemannian metric ds 2 on M means a Riemannian metric on the eichmüller space eich(s g ) on which the orbiford fundamental group π 1 (M) acts on (eich(s g ), ds 2 ) by isometries. It is known that the mapping class group Mod(S g ) contains torsion-free subgroups of finite indices (see [FM12]). We can pass to a finite cover M of M such that M is a manifold. It is clear that the fundamental group π 1 (M) is a torsion-free subgroup of Mod(S g ) of finite index. heorem 4.1. Let S g be a closed surface of genus g with g 2 and M be a finite cover of the moduli space M g of S g. hen, for any complete Riemannian metric ds 2 on M, there exists a point p 0 M such that the Ricci tensor at p 0 satisfies Ric (M,ds 2 )(p 0 ) 0. Proof. We argue it by contradiction. Suppose it is not. hat is, there exists a complete Riemannian metric ds 2 on M such that for all p M the Ricci tensor Ric (M,ds 2 )(p) = 0. As described above, if necessary we pass to a finite cover M such that M is a manifold. We lift the metric ds 2 onto M, still denoted by ds 2. hen, (4.1) Ric (M,ds 2 ) (p) = 0. Let α, be two nontrivial simple closed curves on S g with the geometric intersection i(α, ) 2, and τ α, τ be the Dehn-twists along α and respectively. Since the fundamental group π 1 (M) is a subgroup of Mod(S g )

11 SCALAR CURVAURE 11 of finite index, there exists n 0, m 0 Z + such that τ n 0 α, τ m 0 Lemma 2.1 we know that the group (4.2) < τ n 0 α, τ m 0 > = F 2 π 1 (M). From where F 2 is a free group of rank 2. For sure < τ n 0 α, τ m 0 > acts on the universal cover (eich(s g ), ds 2 ) of (M, ds 2 ) by isometries. We endow < τ n 0 α, τ m 0 > with the word metric dist word w.r.t the generator set {τ n 0 α, τ n 0 α, τ m 0, τ m 0 }. Let e be the unit in < τ n 0 α, τ m 0 >. For any r > 0 we set B(e, r) := {φ < τ n 0 α, τ m 0 >: dist word (φ, e) r}. Let q 0 (eich(s g ), ds 2 ) and dist ds 2 be the induced path metric of (eich(s g ), ds 2 ) on eich(s g ). We define (4.3) C := max {dist ds 2(τ n 0 α q 0, q 0 ), dist ds 2(τ m 0 q 0, q 0 )} > 0. he triangle inequality leads to dist ds 2(φ q 0, q 0 ) r C, φ B(e, r). Since < τ n 0 α, τ m 0 > acts freely on the universal cover (eich(s g ), ds 2 ) of (M, ds 2 ), there exists a number ɛ 0 > 0 such that which implies (4.4) dist ds 2(γ q 0, q 0 ) > 2ɛ 0, e γ < τ n 0 α, τ m 0 > γ 1 B(q 0 ; ɛ 0 ) γ 2 B(q 0 ; ɛ 0 ) =, γ 1 γ 2 < τ n 0 α, τ m 0 > where B(q 0 ; ɛ 0 ) := {p (eich(s g ), ds 2 ); dist ds 2(p, q 0 ) ɛ 0 }. From inequality (4.3) and the triangle inequality we know that, for all r > 0, (4.5) γ B(q 0 ; ɛ 0 ) B(q 0 ; r C + ɛ 0 ). γ B(e,r) Equation (4.4) tells that the geodesic balls {γ B(q 0 ; ɛ 0 )} γ B(e,r) are pairwisely disjoint. hus, by taking the volume, equations (4.4) and (4.5) lead to (4.6) γ B(e,r) Vol(γ B(q 0, ɛ 0 )) = Vol( γ B(e,r) γ B(q 0 ; ɛ 0 )) Vol(B(q 0 ; r C + ɛ 0 )). Since the group < τ n 0 α, τ m 0 > acts on (eich(s g ), ds 2 ) by isometries, Vol(γ B(q 0, ɛ 0 )) = Vol(B(q 0 ; ɛ 0 )) for all γ < τ n 0 α, τ m 0 >. From inequality (4.6) we have (4.7) #B(e, r) Vol(B(q 0, ɛ 0 )) Vol(B(q 0, r C + ɛ 0 )).

12 12 KEFENG LIU AND YUNHUI WU (4.8) Rewrite it as #B(e, r) Vol(B(q 0; r C + ɛ 0 )). Vol(B(q 0 ; ɛ 0 )) Since (eich(s g ), ds 2 ) is complete, from equation (4.1) and the Gromov- Bishop volume comparison inequality (see [Gro07]), we have, for all r > 0, (4.9) (4.10) #B(e, r) Vol(B(q 0; r C + ɛ 0 )) Vol(B(q 0 ; ɛ 0 )) (r C + ɛ 0) 6g 6 ɛ 6g 6 0. Which in particular implies that the group < τ n 0 α, τ m 0 > Mod(S g ) has polynomial growth, which contradicts equation (4.2) since the free group F 2 has exponential growth. Remark 4.2. he above argument is standard. One can see [Gro07] for more applications of this argument. Actually one may conclude that any finite cover of the moduli space M g admits no complete Riemannian metric of nonnegative Ricci curvature by using the same argument. 5. Deformation to Positive Scalar curvature As stated in the introduction Farb and Weinberger in [FW] showed that the set of complete Riemannian metrics of positive scalar curvatures on the moduli space M g is not empty. In this section we will show that any complete Riemannian metric of nonnegative scalar curvature on a manifold which finitely covers M g can be deformed to a new complete Riemannian metric of positive scalar curvature which is equivalent to the ambient metric. his will be applied to prove heorem 1.2. In [Kaz82] Kazdan showed that any Riemannian metric of zero scalar curvature on a manifold, whose dimension is greater than or equal to 3, can be deformed to a new metric of positive scalar curvature which is equivalent to the ambient metric provided that the ambient metric is not Ricci flat. Actually his method also works when the scalar curvature is nonnegative. For the sake of completeness we sketch the proof. One can see [Kaz82] for more details. Let M be a finite cover of M g which is a manifold and ds 2 be a complete Riemannian metric on M which has nonnegative scalar curvature. Since the metric is smooth, for any p 0 M there exists a constant r 1 > 0 such that the geodesic ball B(p 0 ; r 1 ) centered at p 0 of radius r 1 has smooth boundary B(p 0 ; r 1 ) and smooth outer normal derivative ν on B(p 0, r 1 ). It suffices to choose r 1 to be less than the injectivity radius of M at p 0.

13 SCALAR CURVAURE 13 We let Sca ds 2 be the scalar curvature of (M, ds 2 ) and ds 2 be the Laplace operator of (M, ds 2 ). Consider the operator 2(6g 7) (5.1) L ds 2(u) := 3g 4 ds 2u + Sca ds 2 u where u C ((M, ds 2 ), R). For 0 < r < inj(p 0 ) where inj(p 0 ) is the injectivity radius of (M, ds 2 ) at p 0. Let µ 1 (B(p 0 ; r)) be the lowest eigenvalue of L with Neumann boundary conditions u ν = 0 on B(p 0; r). It is well-known that (5.2) µ 1 (B(p 0 ; r)) = inf v C ((M,ds 2 ),R) B(p 0 ;r) ( v 2 ds + Sca 2 ds 2 v 2 ) dvol. B(p 0 ;r) v2 dvol he following result is heorem A in [Kaz82] which is crucial in this section. heorem 5.1 (Kazdan). Assume that µ 1 (B(p 0 ; r)) > 0. hen there is a solution u > 0 on (M, ds 2 ) of Lu > 0; in fact there exist two constants C 1, C 2 > 0 such that 0 < C 1 u(p) C 2, p (M, ds 2 ). Assume that u > 0 on (M, ds 2 ). We define the conformal metric (5.3) ds 2 u := u 2 3g 4 ds 2 Direct computation shows that the scalar curvature Sca ds 2 u of ds 2 u is given by the formula (5.4) (5.5) hus, 2(6g 7) L ds 2u = 3g 4 ds 2u + Sca ds 2 u = Sca ds 2 u u 3g 2 3g 4 (5.6) Sca ds 2 u = L ds 2u u 3g 2 3g 4 heorem 5.2. Let S g be a closed surface of genus g with g 2 and M be a finite cover of the moduli space M g of S g such that M is a manifold. hen for any complete Riemannian metric ds 2 of nonnegative scalar curvature on M, there exists a new metric ds 2 1 on M such that (1). he scalar curvature Sca (M,ds 2 1 ) > 0 on (M, ds2 1 ). (2). ds 2 1 ds2. Proof. We follow exactly the same argument as in [Kaz82]. First from heorem 4.1 we know that that there exists a point p 0 M such that the Ricci tensor (5.7) Ric (M,ds 2 )(p 0 ) 0.

14 14 KEFENG LIU AND YUNHUI WU We let r 1 be a constant with 0 < r 1 < inj(p 0 ). Pick a function η C0 (B(p 0, r 1 ); R 0 ) with η(p 0 ) > 0 and consider a family of metrics ds 2 t := ds 2 t η Ric (M,ds 2 ) with scalar curvature Sca (M,ds 2 t ) and the corresponding operator L ds 2 defined t in equation (5.1) with lowest Neumann eigenvalue µ 1 (B(p 0, r 1 ), t). he first variation formula (see [Kaz82]) gives that (5.8) d dt µ 1(B(p 0, r 1 ), t) t=0 = η < Ric, Ric > Vol((B(p 0, r 1 )) where <. > is the standard inner product for tensors in the ds 2 metric. Since η(p 0 ) > 0, equations (5.7) and (5.8) give that d (5.9) dt µ 1(B(p 0, r 1 ), t) t=0 > 0. Since Sca (M,ds 2 ) 0, equation (5.2) gives that µ 1 (B(p 0, r 1 ), 0) = µ 1 (B(p 0, r 1 )) 0. hus, from inequality (5.9) we know that for small enough t 0 > 0, (5.10) µ 1 (B(p 0, r 1 ), t 0 ) > 0 It is clear that (5.11) ds 2 t 0 ds 2. Because of inequality (5.10) we apply heorem 5.1 to (M, ds 2 t 0 ). hus, there is a smooth function u on (M, ds 2 t 0 ) such that (5.12) L ds 2 t0 u(p) > 0 and u(p) > 0, p (M, ds 2 t 0 ). And there exist two constants C 1, C 2 > 0 such that (5.13) 0 < C 1 u(p) C 2, p (M, ds 2 t 0 ). hen we define the new metric as (5.14) ds 2 1 := u 2 3g 4 ds 2 t 0. It is clear that Part (1) follows from equations (5.6) and (5.12). And Part (2) follows from equations (5.11), (5.14) and inequality (5.13). 6. Proof of heorem 1.2 Before we prove heorem 1.2, let us make some preparation and fix the notations. Let (M 1, ds 2 1 ), (M 2, ds 2 2 ) be two Finsler manifolds of the same dimensions, and f : (M 1, ds 2 1 ) (M 2, ds 2 2 ) be a smooth map. For C > 0, we call that f is an C contraction if for any p M 1, (6.1) f (V ) ds 2 2 C V ds 2 1, V p M 1. Recall that the degree deg(f) of f is defined as

15 SCALAR CURVAURE 15 (6.2) deg(f) = where p is a regular value of f. q f 1 (p) sign(det f (q)) Definition 6.1 (Gromov-Lawson). An n-dimensional Riemannian manifold X is called hyperspherical if for every ɛ > 0, there exists an ɛ-contraction map f ɛ : X S n of nonzero degree onto the standard unit n-sphere such that f ɛ is a constant outside a compact subset in X. he following result was proved by Gromov and Lawson in [GL83]. heorem 6.2 (Gromov-Lawson). A complete aspherical Riemannian manifold X cannot have positive scalar curvature if the universal cover X of X is hyperspherical. Proof. See the proof of heorem 6.12 in [GL83]. A direct corollary of heorem 6.2 is heorem 6.3 (Gromov-Lawson). Let (X, ds 2 1 ) be a complete Riemannian manifold of nonpositive sectional curvature. hen for any Riemannian metric ds 2 2 on X with ds2 2 ds2 1, the scalar curvature of (X, ds2 2 ) cannot be positive everywhere on X. We outline the proof of heorem 6.3 as follows, one can see [GL83] or heorem 1.1 in [Roe93] for more details. Outline proof of heorem 6.3. Since ds 2 1 is complete and ds2 2 ds2 1, the manifold (X, ds 2 2 ) is also complete. From heorem 6.2 it suffices to show that the universal cover ( X, ds 2 2 ) of (X, ds2 2 ) is hyperspherical. First since ds 2 2 ds2 1, the identity map i : ( X, ds 2 2) ( X, ds 2 1) is a k-contraction diffeomorphism for some k > 0. Since (X, ds 2 1 ) has nonpositive sectional curvature, the inverse of the exponential map at a point p X exp 1 p : ( X, ds 2 1) R n is a 1-contraction diffeomorphism. Where n = dim(x). It is easy to see that the Euclidean space R n, endowed with the standard metric, is hyperspherical. hat is, for every ɛ > 0, there exists a map F ɛ : R n S n which is an ɛ-contraction of nonzero degree, surjective, and is a constant map outside a compact subset in R n. hen, the composition map F ɛ exp 1 p i : ( X, ds 2 2) S n

16 16 KEFENG LIU AND YUNHUI WU is a k ɛ-contraction of nonzero degree, surjective, and is a constant map outside a compact subset in ( X, ds 2 2 ). Which in particular implies that ( X, ds 2 2 ) is hyperspherical. Now let us state the theorem (heorem 1.2) we are going to prove in this section. heorem 6.4. Let S g be a closed surface of genus g with g 2 and M be a finite cover of the moduli space M g of S g. hen for any Riemannian metric ds 2 on M with ds 2 ds 2, we have inf Sca(p) < 0. p (M,ds 2 ) Recall that in the proof of heorem 6.3, the nonpositivity of the sectional curvature is crucial because in this case the inverse of the exponential map is a contraction. In the setting of heorem 6.4, although the inverse of the exponential map is well-defined by heorem 2.2, it is far from a contraction. In fact, Masur in [Mas75] showed that there exists two different geodesics in (eich(s g ), ds 2 ) starting from the same point such that they have bounded Hausdorff distance. In particular, (eich(s g ), ds 2 ) is not nonpositively curved in the sense of metric spaces. Hence, the argument in [GL83] can not be directly applied to show heorem 6.4. Furthermore the following question is still open. Question 1. Is (eich(s g ), ds 2 ) hyperspherical? We are going to steer clear of Question 1 to prove heorem 6.4. It is very interesting to know the answer to Question 1. Before we prove heorem 6.4, we first provide two important properties for the eichmüller space (eich(s g ), ds 2 ) where ds 2 ds 2, which will be applied later. Definition 6.5. Let X be a metric space. We call X is uniformly contractible if there is a function f : (0, ) (0, ) so that for each x X and r > 0, the ball B(x; r) of radius r centered at x is contractible in the concentric ball B(x; f(r)) of radius f(r). Proposition 6.6. Let ds 2 be a Riemannian metric on eich(s g ) such that ds 2 ds 2. hen (eich(s g), ds 2 ) is uniformly contractible. In particular, (eich(s g ), ds 2 M ), (eich(s g), ds 2 τ ) and (eich(s g ), d 2 LSY ) are uniformly contractible. Proof. From heorem 3.1 and 3.2 we know that ds 2 M ds2 and ds2 τ ds 2 LSY ds2. It suffices to show (eich(s g), ds 2 ) is uniformly contractible provided that ds 2 ds 2. Since ds 2 ds 2, there exist two constants k 1, k 2 > 0 such that k 1 ds 2 ds 2 k 2 ds 2.

17 SCALAR CURVAURE 17 In particular, we have, for each p eich(s g ) and r > 0 (6.3) B ds 2(p; r) B ds 2 (p; r k 1 ) B ds 2(p; k 2 k 1 r) where B ds 2(p; r) := {q eich(s g ); dist ds 2(p, q) r} andb ds 2 (p; r) := {q eich(s g ); dist ds 2 (p, q) r}. Proposition 2.3 tells that the eichmüller ball B ds 2 (p; r) is contractible for all r > 0 and p eich(s g ). hus, equation (6.3) tells that B ds 2(p; r) is contractible in B ds 2(p; k 2 k 1 r). herefore, the conclusion follows by choosing f(r) = k 2 k 1 r. Definition 6.7. Let X be a metric space. We call that X has bounded geometry in the sense of coarse geometry if for every ɛ > 0 and every r > 0, there exists an integer n(r, ɛ) > 0 such that for each x X every ball B(x; r) contains at most n(r, ɛ) ɛ-disjoint points. Where ɛ-disjoint means that any two different points are at at least ɛ distance from each other. Proposition 6.8. Let ds 2 be a Riemannian metric on eich(s g ) such that ds 2 ds 2. hen (eich(s g), ds 2 ) has bounded geometry in the sense of coarse geometry. In particular, (eich(s g ), ds 2 M ), (eich(s g), ds 2 τ ) and (eich(s g ), d 2 LSY ) have bounded geometry in the sense of coarse geometry. Proof. From heorem 3.1 we know that ds 2 M ds2. From Definition 6.7 it suffices to show that (eich(s g ), ds 2 M ) has bounded geometry in the sense of coarse geometry. Let ɛ 0 > 0 be the lower bound for the injectivity radius of (eich(s g ), ds 2 M ) in heorem 3.1. For every r > 0 and every ɛ > 0. Let p eich(s g ) and B ds 2 (p; r) := {q eich(s M g ); dist ds 2 (p, q) r} be the geodesic ball of M radius r centered at p. Assume K = {x i } k i=1 be an arbitrary ɛ- disjoint points in B ds 2 (p; r). hat is M (6.4) dist ds 2 M (x i, x j ) ɛ, 1 i j k. Let ɛ 1 = min { ɛ 4, ɛ 0}. First the triangle inequality tells that (6.5) k i=1b(x i ; ɛ 1 ) B(p, r + ɛ 1 ) B(p, r + ɛ 0 ). By our assumptions that ɛ 1 ɛ 4, inequality (6.4) gives that (6.6) B(x i ; ɛ 1 ) B(x j ; ɛ 1 ) =, 1 i j k. (6.7) From equations (6.5) and (6.6) we have k Vol(B(x i ; ɛ 1 )) Vol(B(p, r + ɛ 0 )). i=1

18 18 KEFENG LIU AND YUNHUI WU From heorem 3.1 we know that the sectional curvatures of (eich(s g ), ds 2 M ) have a lower bound, which, by using the Gromov-Bishop volume comparison inequality, in particular implies that there exists a constant C(r, ɛ 0, g) > 0 depending on r, ɛ 0 and the genus g such that the volume (6.8) Vol(B(p, r + ɛ 0 )) C(r, ɛ 0, g). On the other hand, from heorem 3.1 we know that the sectional curvatures of (eich(s g ), ds 2 M ) have a upper bound. Since ɛ 1 ɛ 0 = inj(eich(s g ), ds 2 M), Elementary Riemannian geometry tells that there exists a constant D(ɛ 1, g) > 0 depending on ɛ 1 and the genus g such that the volume (6.9) Vol(B(x i, ɛ 1 )) D(ɛ 1, g) > 0, 1 i k. Inequalities (6.7), (6.8) and (6.9) give that (6.10) k C(r, ɛ 0, g) D(ɛ 1, g). hen the conclusion follows by choosing n(r, ɛ) = C(r, ɛ 0, g) D(ɛ 1, g). he following result of A. N. Dranishnikov in [Dra03] will be applied to prove heorem 6.4. heorem 6.9 (Dranishnikov). Let X be a complete uniformly contractible Riemannian manifold with bounded geometry whose asymptotic dimension is finite, then the product X R n, endowed with the product metric, is hyperspherical for some positive number n Z. We remark here that the statement of the theorem above is different from heorem 5 (or heorem B) in [Dra03] where there is no condition on the bounded geometry. But if one checks the proof of heorem 5 in [Dra03], heorem 5 follows from heorem 4 and Lemma 4 in [Dra03] where heorem 4 requires that the space X has bounded geometry. We are grateful to Prof. Dranishnikov for the clarification. Now we are ready to prove heorem 1.2. Proof of heorem 1.2. Let M be a finite cover of the moduli space M g of S g and ds 2 be a Riemannian metric on M such that ds 2 ds 2. hat is, there exists a constant k 1 > 0 such that (6.11) ds 2 k 1 d 2. We argue it by contradiction. Assume that (6.12) inf Sca(p) 0. p (M,ds 2 )

19 SCALAR CURVAURE 19 If necessary, we pass to a finite cover of M, still denoted by M, such that M is a manifold. From inequality (6.11) we know that (M, ds 2 ) is complete since the eichmüller metric is complete. hus, from heorem 5.2 we know that there exists a new metric ds 2 1 on M such that (6.13) Sca(p) > 0, p (M, ds 2 1) and (6.14) ds 2 1 ds 2. Let ds 2 M be the McMullen metric on M. In fact the Ricci metric or the perturbed Ricci metric also works here. From Proposition 6.6, Proposition 6.8 and heorem 3.4 we know that the universal cover (eich(s g ), ds 2 M ) of (M, ds 2 M ) is uniformly contractible, has bounded geometry and asydim((eich(s g ), ds 2 M)) <. Hence, one may apply heorem 6.9 to get a positive integer n such that (eich(s g ), ds 2 M ) Rn, endowed with the product metric, is hyperspherical. We pick this integer n Z + and consider the product space (eich(s g ), ds 2 1) R n where (eich(s g ), ds 2 1 ) is the universal cover of (M, ds2 1 ). It is clear that (eich(s g ), ds 2 1 ) Rn is a complete (6g 6+n)-dimensional Riemannian manifold, and the scalar curvature of (eich(s g ), ds 2 1 ) Rn satisfies that (6.15) Sca((p, v)) = Sca(p) > 0 where (p, v) is arbitrary in (eich(s g ), ds 2 1 ) Rn. Claim: he complete product manifold (eich(s g ), ds 2 1 ) Rn is hyperspherical. Proof of the Claim: First since ds 2 1 ds2 (see equation (6.14)), the identity map (6.16) i 1 : (eich(s g ), ds 2 1) R n (eich(s g ), ds 2 ) R n. is a c 1 -contraction diffeomorphism for some constant c 1 1. Since we assume that ds 2 ds 2 (by assumption), the identity map (6.17) i 2 : (eich(s g ), ds 2 ) R n (eich(s g ), ds 2 ) R n. is a c 2 -contraction diffeomorphism for some constant c 2 1. From heorem 3.1 we know that ds 2 ds2 M. hus, the identity map (6.18) i 3 : (eich(s g ), ds 2 ) R n (eich(s g ), ds 2 M) R n. is a c 3 -contraction diffeomorphism for some constant c 3 1.

20 20 KEFENG LIU AND YUNHUI WU By the choice of n Z + we know that for every ɛ > 0 there exists an ɛ-contraction map (6.19) f ɛ : (eich(s g ), ds 2 M) R n S 6g 6+n. such that f ɛ is of nonzero degree onto the unit (6g 6 + n)-sphere and f ɛ is a constant outside a compact subset in (eich(s g ), ds 2 M ) Rn. Consider the following composition map (6.20) F ɛ : (eich(s g ), ds 2 1) R n S 6g 6+n (p, v) f ɛ i 3 i 2 i 1 (p, v) where (p, v) is arbitrary in (eich(s g ), ds 2 1 ) Rn. Since i 1, i 2 and i 3 are diffeomorphisms and f ɛ has nonzero degree, F ɛ also has nonzero degree by the definition. Since i 1, i 2 and i 3 are diffeomorphisms and f ɛ is a constant outside a compact subset of (eich(s g ), ds 2 M ) Rn, a standard argument in set-point topology gives that F ɛ is also a constant outside a compact subset of (eich(s g ), ds 2 M ) R n. It is clear that F ɛ is onto because i 1, i 2, i 3 and f ɛ are onto. It remains to show that F ɛ is a contraction. For every point (p, v) (eich(s g ), ds 2 1 ) Rn and any tangent vector W (p,v) ((eich(s g ), ds 2 1 ) R n ) = R 6g 6+n, (F ɛ ) (W ) = (f ɛ i 3 i 2 i 1 ) (W ) ɛ (i 3 i 2 i 1 ) (W ) ɛ c 3 (i 2 i 1 ) (W ) ɛ c 3 c 2 (i 1 ) (W ) ɛ c 3 c 2 c 1 W where is the standard Euclidean norm in R 6g 6+n. Since ɛ > 0 is arbitrary and c 1, c 2, c 3 > 0, the claim follows. From the claim above and heorem 6.2 of Gromov-Lawson we know that the product manifold (eich(s g ), ds 2 1 ) Rn cannot have positive scalar curvature which contradicts inequality (6.15). Remark he following more general statement follows from exactly the same argument as the proof of heorem 1.2. heorem Let S g be a closed surface of genus g with g 2 and M be any cover which may be an infinite cover, of the moduli space M g of S g such that the orbiford fundamental group of M contains a free subgroup of rank 2. hen for any Riemannian metric ds 2 on M with ds 2 ds 2 we have inf Sca(p) < 0. p (M,ds 2 )

21 SCALAR CURVAURE Proof of heorem 1.5 While heorem 6.3 of Gromov-Lawson tells that any locally symmetric space Γ\G/K admits no Riemannian metric of positive scalar curvature such that the metric is equivalent to the canonical metric on Γ\G/K, Block and Weinberger showed in [BW99] that Γ\G/K admits a complete Riemannian metric of uniformly positive scalar curvature if and only if Γ is an arithmetic group of Q-rank 3. Analogous to Block-Weinberger s theorem, Farb and Weinberger recently announced a result which states that a finite cover of the moduli space M g admits a complete, finite volume Riemannian metric of uniformly positive scalar curvature if and only if g 2. See heorem 4.5 in [Far06]. Both metrics of positive scalar curvature in [BW99, Far06] has the quasiisometry of rays. Chang in [Cha01] showed that any locally symmetric space Γ\G/K admits no Riemannian metric of positive scalar curvature such that the metric is quasi-isometric to the canonical metric on Γ\G/K. Analogous to Chang s result, Farb-Weinberger posed Conjecture 1.3. One can also see Conjecture 4.6 in [Far06]). In this section we will prove our result which implies this conjecture. Definition 7.1. Let M be a cover, which may be an infinite cover, of the moduli space M g and ds 2 be a Riemannian metric on M. We call that ds 2 is quasi-isometric to the eichmüller metric ds 2 if there exist two positive constants L 1 and K 0 such that on the universal cover (eich(s g ), ds 2 ) ((eich(s g ), ds 2 )) of (M, ds2 ) ((M, ds 2 )) respectively, the identity map satisfies L 1 dist ds 2 (p, q) K dist ds 2(p, q) L dist ds 2 (p, q)+k, p, q eich(s g ). If K = 0, ds 2 is equivalent to ds 2. In the quasi-isometry setting, the identity map, defined in the equation (6.17) in the proof of heorem 1.2, may not be a contraction. herefore, the proof of heorem 1.2 can not directly lead to heorem 1.5. Instead of applying heorem 6.2 in the proof of heorem 1.2, we apply the following theorem of Yu in [Yu98] to prove heorem 1.5. One can see Corollary 7.3 in [Yu98]. heorem 7.2 (Yu). A uniformly contractible Riemannian manifold with finite asymptotic dimension cannot have uniform positive scalar curvature. Now we are ready to prove heorem 1.5. Proof of heorem 1.5. Let M be a cover of the moduli space M g of S g and ds 2 be a Riemannian metric on M such that ds 2 is quasi-isometric to ds 2. hat is, there exist two constants L 1 and K > 0 such that (7.1) L 1 dist ds 2 (p, q) K dist ds 2(p, q) L dist ds 2 (p, q)+k, p, q eich(s g ).

22 22 KEFENG LIU AND YUNHUI WU heorem 3.4 gives that (7.2) asydim((eich(s g ), ds 2 )) <. From heorem 7.2 of Yu, it remains to show that (eich(s g ), ds 2 ) is uniformly contractible. We follow a similar argument in the proof of Proposition 6.6 to finish the proof. For each p eich(s g ) and every r > 0, inequality (7.1) and the triangle inequality lead to (7.3) B ds 2(p; r) B ds 2 (p; L (r + K)) B ds 2(p; L 2 (r + K) + K) where B ds 2(p; r) := {q eich(s g ); dist ds 2(p, q) r} andb ds 2 (p; r) := {q eich(s g ); dist ds 2 (p, q) r}. Proposition 2.3 tells that the eichmüller ball B ds 2 (p; L (r + K)) is contractible for all r > 0 and p eich(s g ). hus, equation (7.3) tells that B ds 2(p; r) is contractible in B ds 2(p; L 2 (r + K) + K). hus, the conclusion follows by choosing f(r) = L 2 (r + K) + K. Remark 7.3. heorem 1.5 also holds in the following sense of quasi-isometry, where we call that ds 2 is quasi-isometric to the eichmüller metric ds 2 if there exist two positive constants L 1, K 0 and a map such that for all p, q eich(s g ), f : (eich(s g ), ds 2 ) (eich(s g ), ds 2 ) L 1 dist ds 2 (p, q) K dist ds 2(f(p), f(q)) L dist ds 2 (p, q) + K. If we assume that (eich(s g ), ds 2 ) is quasi-isometric to (eich(s g ), ds 2 ), then the space (eich(s g ), ds 2 ) is also quasi-isometric to (eich(s g ), ds 2 M ) or (eich(s g ), ds 2 LSY ), where these two metrics are uniformly contractible. Indeed, heorem 7.1 in [Yu98] and heorem 3.4 give that the coarse Baum- Connes conjecture holds for (eich(s g ), ds 2 M ) or (eich(s g), d 2 LSY ). hen one can see Corollary 3.9 in [Roe96] and use the same argument in the proof of heorem 1.5 to get the conclusion. 8. One question Let N be an n-dimensional complete simply-connected Riemannian manifold of nonpositive sectional curvature. For any p N, let p N be the tangent space of N at p, it is well-known that the inverse of the exponential map at p exp 1 p : N p N = R n is a 1-contraction diffeomorphism. And this property plays an important role in the proof of heorem 6.3 of Gromov-Lawson. he following question arises in this project.

23 Question 2. Is there any proper differential map SCALAR CURVAURE 23 F : (eich(s g ), ds 2 ) R 6g 6 such that F is a 1-contraction of degree one? diffeomorphism? Moreover, could F be a he constant 1 for the contraction property in this question is not essential, since one can take a rescaling on the target space R 6g 6. An affirmative answer to Question 2 will give another proof of heorem 1.2 by following exact the same argument in [GL83]. See heorem 6.2. We enclose this section by recalling several well-known parametrizations which may be helpful for this question. (1). Recall that the eichmüller parametrization at a point X eich(s g ) is given by F X : (eich(s g ), ds 2 ) R 6g 6 = S 6g 7 R 0 Y (V [X, Y ], dist (X, Y )) where V [X, Y ] is the direction of the eichmüller geodesic from X to Y. It is well-known that F X is a proper differential map of degree one. However, F X is not a contraction since there exists two geodesics starting at X which have bounded Hausdorff distance, which was proved by Masur in [Mas75]. (2). Let X eich(s g ) be a hyperbolic surface and QD(X) be the holomorphic quadratic differential on X which can be identified with R 6g 6. Let X be the Bers embedding of (eich(s g ), ds 2 ) into QD(X) with respect to the base point X. It is well-known that X is a contraction (For example one can see heorem 4.3 in [FKM13]). However, X is not proper since the image of the Bers embedding is a bounded subset in R 6g 6. (3). Fix a hyperbolic surface X eich(s g ). hen for any Y eich(s g ) there exists a unique harmonic map from X to Y which is isotopic to the identity map. he Hopf differential F X (Y ) of this harmonic map is a holomorphic quadratic differential on X. In particular this gives a differential map from eich(s g ) to QD(X). Wolf in [Wol89] showed that this map F X : (eich(s g ), ds 2 ) QD(X) = R 6g 6 is a diffeomorphism. In particular, this map is proper of degree one. However, Markovic in [Mar02] showed that F X is not a contraction (One can see heorem 2.2 in [Mar02]). (4). Since ds 2 ds2 W P, the identity map i : (eich(s g ), ds 2 ) (eich(s g ), ds 2 W P )

24 24 KEFENG LIU AND YUNHUI WU is a contraction diffeomorphism. Fix a hyperbolic surface X eich(s g ). Since the sectional curvature of the Weil-Petersson metric is negative [ro86, Wol86] and the Weil-Petersson metric is geodescally convex [Wol87], the inverse of the exponential map at X exp 1 X : eich(s g) X (eich(s g )) = R 6g 6 is a 1-contraction. Consider the composition map F X : (eich(s g ), ds 2 ) X (eich(s g )) = R 6g 6 Y exp 1 X i(y ) It is not hard to see that F X is a contraction and differential map of degree one. However, since the Weil-Petersson metric is incomplete [Chu76, Wol75], F X is not proper. References [Ahl61] Lars V. Ahlfors, Some remarks on eichmüller s space of Riemann surfaces, Ann. of Math. (2) 74 (1961), [BBF14] Mladen Bestvina, Ken Bromberg, and Koji Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publications mathmatiques de l IHS (2014), [BD08] G. Bell and A. Dranishnikov, Asymptotic dimension, opology Appl. 155 (2008), no. 12, [BF06] Jeffrey Brock and Benson Farb, Curvature and rank of eichmüller space, Amer. J. Math. 128 (2006), no. 1, [BW99] Jonathan Block and Shmuel Weinberger, Arithmetic manifolds of positive scalar curvature, J. Differential Geom. 52 (1999), no. 2, [Cha01] Stanley S. Chang, Coarse obstructions to positive scalar curvature in noncompact arithmetic manifolds, J. Differential Geom. 57 (2001), no. 1, [Chu76] ienchen Chu, he Weil-Petersson metric in the moduli space, Chinese J. Math. 4 (1976), no. 2, [Dra03] A. N. Dranishnikov, On hypersphericity of manifolds with finite asymptotic dimension, rans. Amer. Math. Soc. 355 (2003), no. 1, [Far06] Benson Farb, Some problems on mapping class groups and moduli space, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp [FKM13] Alastair Fletcher, Jeremy Kahn, and Vladimir Markovic, he moduli space of Riemann surfaces of large genus, Geom. Funct. Anal. 23 (2013), no. 3, [FM10] Benson Farb and Howard Masur, eichmüller geometry of moduli space. II. M(S) seen from far away, In the tradition of Ahlfors-Bers. V, Contemp. Math., vol. 510, Amer. Math. Soc., Providence, RI, 2010, pp [FM12] Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, [FW] Benson Farb and Shmuel Weinberger, Positive scalar curvature metrics on the moduli space of Riemann surfaces, In preparation. [GL80] Mikhael Gromov and H. Blaine Lawson, Jr., he classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, [GL83], Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. (1983), no. 58, (1984).