Compactification of a class of conformally flat 4-manifold

COMPACTIFICATION OF A CLASS OF CONFORMALLY FLAT 4-MANIFOLD Sun-Yung A. Chang, Jie Qing and Paul C. Yang Abstra t. In this paper we generalize Huber's result on omplete surfa es of nite total urvature. For omplete lo ally onformally at 4-manifolds of positive s alar urvature with Q urvature integrable, where Q is a variant of the Chern-GaussBonnet integrand; we rst derive the Cohn-Vossen inequality. We then establish niteness of the topology. This allows us to provide onformal ompa ti ation of su h manifolds. S0. Introdu tion In this paper we study the ends of lo ally onformally at 4-dimensional manifolds. Re all in the theory of omplete surfa es of nite total urvature, CohnVossen [CV℄ showed that if the Gauss urvature of a omplete analyti metri is absolutely integrable then (0.1) Z KdA  2; where  is the Euler number of the surfa e. Huber ([H℄) extended this inequality to metri s with weaker regularity and proved that su h surfa e an be onformally ompa ti ed by adjoining a nite number of points. For su h surfa es the de it in formula (0.1) has an interpretation as an isoperimetri onstant. One may represent ea h end onformally as R2 n D for some ompa t set D and onsider the following isoperimetri ratio: L2 (r)  = rlim !1 4A(r) where L(r) is the length of the boundary ir le Br = fjxj = rg, and A(r) the area of the annular region B (r) n D. For a fairly large lass of omplete surfa es alled surfa es with normal metri s, Finn ([F℄) showed that, Z X 1 (0.2) (M ) KdvM = j ; 2 M 0 Resear h of Chang is supported in part by NSF Grant DMS-9706864 and a Guggenheim Foundation Fellowship. 0 Resear h of Qing is supported in part by NSF Grant DMS-9803399 and DMS-9706864. 0 Resear h of Yang is supported in part by NSF Grant DMS-9706507. 1 Typeset by AMS-TEX where the sum is taken over ea h end of M . In dimension four, Greene-Wu ([GW℄) obtained a generaization of (0.1) to omplete manifolds of positive se tional urvature outside a ompa t set. In a previous paper we onsidered a generaliztion of (0.2) in R 4 . In order to des ribe the result we brie y re all the fourth order urvature invariant Q. For onformal geometry in dimension four, the Paneitz operator 2 P =
2 + Æ ( Rg 2Ri )d; 3 where Æ denote the divergen e, and d the di erential, and R is the s alar urvature and Ri the Ri i tensor, plays the same role as the Lapla ian in dimension two ( f. [P℄ [BCY℄ [CY℄, for example). Under onformal hange of metri g = e2w g0 , the Paneitz operator transforms by Pg = e 4w Pg0 . The Paneitz operator de nes a natural fourth order urvature invariant Q: for the onformal metri g = e2w g0 , (0.3) Pg0 w + 2Qg0 = 2Qg e4w where 1 1 (0.4) Q = (
R + R2 3jE j2) 12 4 and E is the tra eless Ri i tensor. The Q urvature invariant is related to the Chern-Gauss-Bonnet integral in dimension four: Z 1 jW j2 + Q)dV (0.5) (M ) = 2 ( 4 M 8 where W is the Weyl tensor and M is a ompa t, losed 4-manifold. We note here the onformal invarian e of the integrand: jW j2 dV remains the same when the metri g undergoes a onformal hange g 0 = e2w g . More generally, when the manifold has a boundary, Chang and Qing [CQ℄ has de ned a boundary operator P3 and its asso iated boundary urvature invariant T : (0.6) P3 w + Tg0 = Tg e3w ; where 1 1 1 3 1~ 1 N R + RH Ranbn Lab + H 3 trL +
H; T= 12 6 9 3 3 H is the mean urvature of the boundary, Lab denotes the se ond fundamental ~ denotes the form of the boundary, N denotes the unit inward normal derivative,
boundary Lapla ian. Then the Chern-Gauss-Bonnet integral is supplemented by Z Z 1 j W j2 1 (0.7) (M ) = 2 ( + Q)dV + 2 (L + L5 + T )d; 4 M 8 4 M 4 where 1 L4 = Rijij Laa + RaNaN Lbb RaNbN Lab + Ra b Lab 3 and 2 L5 = Laa Lbb L + Laa Lb Lb Lab Lb L a 9 where i; j = 1; 2; 3; 4, a; b; = 1; 2; 3, and N is the inward normal dire tion. In analogy with the Weyl term, (L4 + L5 )d is a pointwise onformal invariant. 2 Theorem [CQY℄. Suppose that e2w jdxj2 on R4 is a omplete metri with its Qurvature absolutely integrable, and suppose that its s alar urvature is nonnegative at in nity. Then (0.8) 1 Z 4 1 4w dx = lim (vol(Br (0))) 3 Qe r!1 4(2 2 ) 31 vol(Br (0)) 4 2 R4  0: In this paper we extend this result to more general situations. First we lo alize arguments in ([CQY℄) to an end orresponding to a pun ture and obtain: Theorem 1. Suppose (R 4 n B; e2w jdxj2) is a omplete onformal metri with nonnegative s alar urvature at in nity. If in addition Z then jQjdV < 1; Z 4 Z (vol(Br )) 3 1 1 3w lim T e Qe4w dx: = 1 2 r!1 4(2 2) 3 vol(Br n B ) 4 B 4 2 R4nB This formula makes it possible to extend the basi result (0.8) to allow the domain to have a nite number of pun tures. Corollary. Suppose that (M; g ) is a omplete 4-manifold with only nitely many onformally at simple ends. And suppose that the s alar urvature is nonnegative at ea h end, and the Q urvature is absolutely integrable. Then (M ) k X 1 2 fjW j + 8QgdvM = i ; 32 2 M i=1 Z where in the inverted oordinates entered at ea h end, 4 (vol(Br )) 3 i = rlim !1 4(2 2) 31 vol(Br n B ) : We refer to de nition 1.1 in se tion 1 for the de nition of onformally at simple end. We then use the Chern-Gauss-Bonnet formula to derive a ompa ti ation riteria in analogy with Huber's two dimensional result. Theorem 2. Let (  S 4 ; g = e2w g0 ) be a omplete onformal metri satisfying a) The s alar urvature is bounded between two positive onstants, and jrg Rj is bounded, (b) The Ri i urvature of the metri g has a lower bound, ( ) the Paneitz urvature is absolutely integrable, i.e. Z jQjdvg < 1: 3 then = S 4 n fp1 ; :::; pk g. An essential ingredient in the above niteness result is to view the boundary integral as measuring the growth of volume. The niteness of the Q integral implies a ontrol on the growth of volume, whi h an only a ommodate the growth of a nite number of ends. As a onsequen e we an lassify solutions of the equation Q = onstant in the following Corollary. If (  S 4 ; e2w g0 ) is a omplete onformal metri satisfying onditions (a) (b) ( ) of Theorem 2 and in addition Q is onstant , then there are only two possibilities: 1. (  S 4 ; e2w g0 ) = (S 4 ; g0) orresponding to Q = 3; j2 ) orresponding to Q = 0. 2. (  S 4 ; e2w g0 ) = (R 4 n f0g; jjdx xj2 To provide a more general ontext for this result, we re all that in a study of lo ally onformally at stru tures with positive s alar urvature, S hoen-Yau ([SY℄) proved that the holonomy over of su h manifolds an be onformally embedded as domains into spheres with boundary of small Hausdor dimension. Thus our ompa ti ation riteria applies immediately to simply onne ted manifolds for whi h the onditions (a) (b) and ( ) hold. In fa t, sin e the argument lo alizes to ea h end, we an dispense with simple onne tivity assumption provided we are willing to assume the natural notion of geometri niteness in Kleinian groups. For onvenien e we will follow the terminology of ([Ra℄). Re all that a dis rete group of onformal transformations of the sphere also a ts as hyperboli isometri s on the interior of the sphere. There is a notion of geometri niteness of a Kleinian group that we will state more pre isely in se tion three. Thus we prove Theorem 3. Suppose (M 4 ; g ) is a lo ally onformally at omplete 4-manifold satisfying onditions (a), (b), ( ) and (d) the holonomy representation of the fundamental group is a geometri ally nite Kleinian group without torsion; then M = M n fp1 ; :::; pk g where M is a ompa t lo ally onformally at 4-manifold. We outline the rest of the paper. In se tion one we derive the generalized ChernGauss-Bonnet formula of Theorem 1 and its extensions to more general situations when we only require the ends be lo ally onformally at. In se tion two we study the simply onne ted ase and prove Theorem 2 and its orollary. Finally in se tion three we extend the argument to the non-simply onne ted situation and prove the ompa ti ation result in Theorem 3. A knowledgment: Part of this work was done while Qing and Yang were visiting Prin eton University. They would like to thank the Department of Mathemati s at Prin eton University for support and hospitality. We would also like to thank Feng Luo and Peter Sarnak for their interest and informative onversation. x1. Chern-Gauss-Bonnet Integral In this se tion, we will prove the generalized Chern-Gauss-Bonnet formula for omplete 4-manifolds with only a nite number of onformally at simple ends. 4 More pre isely, we will establish Theorem 1.2 below, whi h generalizes Finn's result [F℄ in two dimension. We rst de ne manifolds with onformally at simple ends. De nition 1.1. Suppose that (M; g ) is a omplete non ompa t 4-manifold su h that M =N k [ [ f Ei g i=1 where (N; g ) is a ompa t Riemannian manifold with boundary N = k [ i=1 Ei ; and ea h Ei is a onformally at simple end of M ; that is: (Ei ; g ) = (R4 n B; e2wi jdxj2 ); for some fun tion wi , where B is the unit ball in R4 . Then we say (M; g ) is a omplete 4-manifold with a nite number of onformally at simple ends. There are many examples of omplete 4-manifolds with only nite number of onformally at simple ends. For example those onstru ted by S hoen [S℄ and Mazzeo-Pa ard [MP℄ on the 4-sphere with a nite number of points deleted and having a omplete, onformally at metri with onstant s alar urvature. Another large set of examples is given by Theorem 3.1 in this paper{they in lude all lo ally onformally at metri s satisfying onditions (a), (b), ( ) and (d) in the statement of Theorem 3.1. In this se tion, we will prove the following result: Theorem 1.2. Suppose that (M; g ) is a omplete 4-manifold with nite number of onformally at simple ends. And suppose that (a) The s alar urvature is non-negative at in nity at ea h end. ( ) Its Q urvature is integrable; that is Z M Then (M ) where jQjdvM < 1: k X 1 2 fjW j + 8QgdvM = i ; 32 2 M i=1 Z R ( Br (0) e3wi d (x)) 3 i = rlim !1 4(2 2) 13 RB (0)nB e4wi dx : r (1.1) 4 5 (1.2) Corollary 1.3. Under the assumptions of Theorem 1.2 we have in parti ular Z 1 (M )  (jW j2 + 8Q)dvM : (1.3) 32 2 M Combining the result in Corollary 1.3 and Theorem 3.1 in se tion 3, we have the following result. Corollary 1.4. Suppose (M; g ) is a lo ally onformally at 4-manifolds whi h satis es onditions (a),(b),( ) and (d) as in the statement of Theorem 3.1; then (M; g ) is onformally equivalent to (M n fpi gki=1 ; e2w jdxj2 ) where M is a ompa t lo ally onformally at 4-manifold and on lusions (1.1), (1.2) hold. The proof we shall present below is a modi ation and sharpened version of the proof in [CQY℄. Thus we will sometimes refer to [CQY℄ for some analyti details. There are three main steps in the proof. First we will establish the theorem (in Proposition 1.6) for manifold with a simple end whi h has an axis of symmetry. We then establish the theorem for metri with a \normal" end (as de ned in de nition 1.7 below) in Proposition 1.11 by omparing the metri with its averaged metri whi h is rotationally symmetri ; nally we prove in Proposition 1.12 that under the assumptions (a) and ( ) of Theorem 1.2, the metri is \normal". To start with, onsider a Riemannian manifold E = (R4 n B; e2w jdxj2 ), where B is 4 the unit ball entered at the origin in R4 , with w a radial fun tion on asR R n B . We sume that the Paneitz urvature Q is absolutely integrable on E , i.e. R4 nB jQje4w dx < 1: In ylindri al oordinates jxj = r = et , we have ( 2 t2  2  2 )( 2 + 2 )v = 2Qe4v ; 0  t < 1; t t t (1.4) where v = w + t. For onvenien e, we denote 2Qe4v by F . Equation (1.4) is equivalent to the following ODE v 0000 4v 00 = F; 0  t < 1 (1.5) R 1 with initial onditions v (0); v 0 (0); v 00 (0); v 000 (0) given, where 0 jF jdt < 1. By the method of variation of oeÆ ients we rst obtain one parti ular solution f to (1.5) as follows: denote f 00 = C (t)e 2t , then C (t) satis es C 00 (t) 4C 0 (t) = F (t)e2t ; or equivalently (C 0 (t)e4t )0 = F (t)e 2t : Thus we an solve for C (t) as: Z 1 Z t 4 x F (y )e 2y dydx e C (t) = x 1 Z Z 1 1 4t 1 t 2 x = e F (x)e2x dx: F (x)e dx 4 4 1 t 6 (1.6) Therefore Z 1 2t 1 00 F (x)e 2x dx f (t) = e 4 t and Z 1 2t t F (x)e2x dx; e 4 0 Z 1 Z 1 2t t 0 2 x 2 t F (x)e 2x dx f (t) = fe F (x)e dx e 8 t 0 Z 1 Z t + F (x)dx F (x)dxg: t (1.7) (1.8) 0 One may determine f by hoosing f (0) = 0. In general v (t) = 0 + 1t + 2e 2t + 2t 3 e + f (t) for some onstants 0 ; 1 ; 2 ; 3 depending on the given initial data of v . The following simple fa ts are proved in [CQY-Lemma 2.1-2.5℄. Lemma 1.5. Z t 2 t F (x)e2x dx = 0: lim e t!1 0 0 (t) = 1 lim f t!1 8 00 lim f (t) = 0 t!1 lim f 000 (t) = 0: t!1 Z 1 1 (1.9) (1.10) F (x)dx (1.11) Suppose, in addition, that either the s alar urvature is nonnegative at in nity or v 00 (t) = O(1) as t ! 1. Then 3 = 0. Thus we may on lude, Proposition 1.6. Suppose that w isRa radial fun tion on R4 n B and e2w jdxj2 is a metri omplete at in nity with R4 nB jQje4w < 1 and its s alar urvature nonnegative at in nity. Then Z 0 (t) = 1 lim v T e3w t!1 4 2 B Z 1 2Qe4w  0: 8 2 R4 nB (1.12) Moreover, we have 4 Z (vol(Br )) 3 1 T e3w lim = 1 2 r!1 4(2 ) 3 vol(Br n B ) 4 2 B Z 1 2Qe4w : 8 2 R4 nB (1.13) Proof. Under the assumptions, we have v (t) = 0 + 1t + 2e 7 2t + f (t): (1.14) Thus, by Lemma 1.5, we have 000 lim v 000 (t) = tlim !1 f (t) = 0: t!1 On the other hand, the Chern-Gauss-Bonnet formula says: Z Z Z 1 1 1 4 v 3 v 2Qe + 2 Te T e3v = 0; 8 2 0ts 4 t=s 4 2 t=0 (1.15) where, as de ned in [CQ, remark 3.1℄, in the spe ial ase of the standard S 3  [0; s℄, we have 1 (1.16) T e3v = P3 v = v 000 + 2v 0 : 2 Therefore Z Z 1 000 1 1 4 v 0 2Qe = ( v (s) + 4v (s)) T e3v : (1.17) 2 2 8 0ts 4 4 t=0 Thus, Z Z 1 1 lim v 0 (t) = 2 T e3w 2Qe4w : 2 t!1 4 B 8 R4 nB From the ompleteness of e2w g0 at in nity, we then on lude that limt!1 v 0 (t)  0. To see (1.13), we write V3 (r) = vol(Br ) = and V4 (r) = vol(Br n B ) = Z Br e3w(r) d; Z rZ 1 Bs e4w(s) dds: If the limit limt!1 v 0 (t) is stri tly positive, then both V4 (r) and V3 (r) tend to in nity as r tends to in nity; then by L'Hospital's rule we have: 4 V33 4jB j 3 r3 e4w (rw0 + 1) lim = rlim r!1 V4 !1 jB jr3e4w 1 0 = 4jB j 3 tlim !1 v (t); 4 thus (1.13) holds. On the other hand, if limt!1 v 0 (t) = 0 and limr!1 V4 (r) is nite, then limt!1 e4v = 0. Hen e limr!1 V3 (r) = 0. (1.13) still holds. We have thus nished the proof of the proposition. We now restri t our attention to the study of a general onformally at simple end, namely, E = (R4 n B; e2w jdxj2) where w is a smooth fun tion on R4 n B . Following Finn [F℄ we de ne the notion of a normal metri . We will show that, for a omplete normal metri on R4 n B , Proposition 1.6 still holds. Suppose that e2w jdxj2 is a metri on R4 n B with its Paneitz urvature Q absolutely integrable on (R4 n B; e2w g0 ), i.e. ondition ( ) holds. 8 De nition 1.7. A onformal metri e2w jdxj2 satisfying ondition ( ) is said to be normal on E = (R4 n B; e2w g0 ) if Z jyj 2Q(y)e4w(y)dy + log jxj + h(x) 1 log (1.18) w(x) = 2 8 R4 nB jx y j where is some onstant and h( jxxj2 ) is some biharmoni fun tion on B . Observe that in general if we all the integral in (1.18) the potential of
2 w, it di ers from w by a biharmoni fun tion. Thus the normal ondition is meant to ontrol the growth of the di eren e between w and the potential of
2 w. We will show later in this se tion that a fairly large lass of metri s on E are normal in parti ular those satis es ondition ( ) and the ondition that the s alar urvature is nonnegative at in nity We ompare a normal metri to its logarithmi average over spheres. The following is a key te hni al lemma omparing the growth of V3 (r) and V4 (r) of a normal metri to those of its averaged metri . Sin e Lemma 1.8 below an be proved following the same outline as the orresponding lemma: Lemma 3.1 in [CQY℄, we will skip its proof here. Lemma 1.8. Suppose that the metri e2w jdxj2 on R4 n B is a normal metri . Then V3 (r) = jBr (0)je3w(r)
eo(1) (1.19) and d V4 (r) = jBr (0)je4w(r)
eo(1) ; (1.20) dr R where w(r) = jB1r (0)j Br (0) w(y )d (y ), and o(1) ! 0 as jxj ! 1. R Next we dis uss the metri e2w jdxj2 on R4 n B , where w(r) = jB1r (0)j Br (0) wd . For onvenien e, we will use the ylindri al oordinates again. Denote by E = (S 3  [0; 1); e2v g ) = (R4 n B; e2w jdxj2 ) (where g is the standard metri of the ylinder S 3  R1 ), then, v satis es Z 1 0000 00 v 4v = 2Qe4w d (y )e4t = F (t); 0  t < 1 jBr (0)j Br (0) with Z 1 Z 1 jF (t)jdt = jS 3j 4 2jQje4w dx < 1: 0 R nB 4 2 w Lemma 1.9. Suppose that (R n B; e jdxj2 ) is a omplete normal metri . Then its averaged metri (R4 n B; e2w jdxj2 ) is also a omplete metri . Proof. This is basi ally a onsequen e of Lemma 1.8. Using the argument of Lemma 1.8 we have Z 1 w(r) w (r) o(1) jS 3j S3 e d = e
e ; where o(1) ! 0 as r ! 1. Hen e Z r1 Z r1 Z Z r1 Z 1 1 w (r) o(1) w ( r ) w ( r ) jS 3j S3 r0 e drd = r0 jS 3j S3 e d dr = r0 e
e dr; whi h proves the Lemma. 9 Lemma 1.10. Suppose that (R4 n B; e2w jdxj2 ) is a normal metri . Then j
w(r)j  rC2 : (1.21) Proof. We ompute Z Z 1 1 w(r )d = 3
w(r) =
3 jS j S 3 jS j S3
w(x) d(x): Sin e the metri is normal, Z Z 1 1 1 1
w(r) = 2 f d (x)g2Q(y )e4w(y)dy + 2 2 +
h: 3 2 8 R4 nB jS j S 3 jx y j jxj Following Finn in [F℄, we have 1 jS 3 j Z 8 > > < 1 2 if jy j < r 1 r 2 d = > 1 S 3 jr y j > : 2 if jy j > r jy j (1.22) sin e jx 1yj2 is the Green's fun tion in 4-D. We also observe that limr!1
h =
h(1) = 0, hen e Z 1 1 j Qje4w dy + 2 2 : 2 2 4 r R4 nB r j
w(r)j  This proves the lemma. Proposition 1.11. Suppose that (R4 n B; e2w jdxj2) is a omplete normal metri . Then R Z ( Br (0) e3w d (x)) 3 1 lim = 2 T e3w R 4w dx r!1 4(2 2) 31 4  e B Br (0)nB 4 Z 1 Qe4w dx  0: 4 2 R4 nB Proof. Write the metri e2w(r) jdxj2 in ylindri al oordinates (S 3  [1; 1); e2v g ) = (R4 n B; e2w g0 ). Then Proposition 1.6 gives Z 0 (t) = 1 lim v  Te3w t!1 4 2 B Z 1  4w dx  0; Qe 4 2 R4 nB where we have used Lemma 1.8 and Lemma 1.9. Noti e that  4w (r) = 2Qe Z Z Z 1 2 1
2 w =
2 w = Qe4w ; jBr (0)j Br (0) jBr (0)j Br (0) jBr (0)j Br (0) 10 whi h implies and that Z Z 1  4w dx = 1 Qe4w dx; Qe 8 2 R4 nB 8 2 R4 nB Te3w = P3 w = P3 w = whi h implies Z 1 3w jBr (0)j Br (0) T e ; Z Z 1 1 Te3w = 2 T e3w : 2 4 B 4 B Therefore we may apply Lemma 1.8 and get R ( Br (0) e3w d (x)) 3 j S 3 j 34 e4w r4 R = rlim lim 4w !1 V4 (r) : r!1 Br (0)nB e dx A ording to (1.20) in Lemma 1.8, the volume of the ylindri al shell Br (0) n B is bounded in the metri e2w jdxj2 if and only if it is bounded in the averaged metri e2w(r) jdxj2 . Thus we may apply argument similar to the proof at the end of Proposition 1.6 to nish the proof of Proposition 1.11 here. Proposition 1.12. Suppose that the Paneitz urvature Q of (R4 n B; e2w jdxj2) is absolutely integrable in (R4 n B; e2w jdxj2 ), and that its s alar urvature is nonnegative at in nity. Then it is a normal metri . Proof. Let Z jyj 2Qe4w dy 3 log (1.24) (x) = 2 4 R4 nB jx y j and = w . We will show that the biharmoni fun tion on R4 n B is of the form log jxj + h(x) for some onstant and some biharmoni fun tion h( jxxj2 ) on B . Re all the transform formula for the s alar urvature fun tion 4
w + jrwj2 = Je2w where J = fra 16R: And noti e that
is a harmoni fun tion on R4 n B . Thus Z 1
(x0) = jBr (x0)j Br (x0 )
d Z Z 1 1 2 = jBr (x0 )j Br (x0 ) (jrwj + J )d jBr (x0 )j Br (x0 )
d (1.25) as long as Br (x0 )  R4 n B . The rst term on the right of (1.25) is ertainly nonpositive sin e J  0 when jx0 j is large enough. For the se ond term, we have Z Z Z 3 1 1 4w(y )dy:
d = 2 f 3 2 d g2Q(y )e 2  j S j j r + x y j 3 4 0 Br (x0 ) S R nB Z 3  22r2 4 2jQje4w dy: R nB 11 Therefore, taking r = 12 jx0 j for jx0 j > 2, for instan e, we have, for any x0 2 R4 n B ,
(x0 )  C jx0j2 for some onstant C depending on . Thus
( + C log jxj)  0: 2 C log jxj, then,
g ( x )  0 and is harmoni on B n f0g. So, if we set g (x) = 2 jxj2 By B^o her's theorem (Theorem 3.9 in [ABR℄), we have
g ( x jxj2 ) = 1 jxj2 + b(x); for some positive onstant and some harmoni fun tion b(x) on B . Hen e g ( jxxj2 )+ x C 1 1 2 log j jxj2 j is a biharmoni fun tion on B . De ne h(x) = + 2 log jxj 2 log jxj, h( jxxj2 ) is biharmoni on B and 1 (x) = ( 2 C ) log jxj + h(x) on R4 n B: We have thus nished the proof of the Proposition. We ombine the results in Proposition 1.7, Proposition 1.11 and Proposition 1.12 to on lude: Corollary 1.12. Suppose that (R4 n B; e2w jdxj2) is a omplete metri with its Paneitz urvature absolutely integrable in (R4 n B; e2w jdxj2 ), and that its s alar urvature is nonnegative at in nity. Then R Z ( Br (0) e3w d (x)) 3 1 T e3u lim = 2 R 4w dx r!1 4(2 2 ) 31 4  e B Br (0)nB 4 Z 1 Qe4u dx  0: 4 2 R4 nB Proof of Theorem 1.2. Suppose M =N k [ [ f Ei g i=1 where (N; g ) is a ompa t Riemannian manifold with boundary N = k [ i=1 12 Ei : Re all the Chern-Gauss-Bonnet formula on the ompa t manifold N k X 1 1 2 (N ) = (jW j + 8Q)dvN + (L + T )dN : 2 2 32 N i=1 4 Ei Sin e ea h end Ei is onformal to R4 n B , we have LjEi = 0 (see [CQ℄). Then, apply Corollary 1.12 to ea h end, we obtain Z Z 1 1 T d = i + 8QdvM : (1.26) 4 2 Ei 32 2 Ei We also observe that the Weyl urvature W vanishes on ea h end Ei , thus from (1.26) we have Z Z k X 1 2 (M ) = (N ) = (jW j + 8Q)dvM + i : 32 2 M i=1 This establishes (1.1). (1.2) is also a dire t onsequen e of Corollary 1.13. We have thus nished the proof of the theorem. Z x2. Simply onne ted ase In this se tion, we will prove the following: Theorem 2.1. Suppose that M is a a subdomain in S 4 and g =  2 g is a omplete onformal metri on M , where g is the standard metri on S 4 ; satisfying: (a) The s alar urvature is bounded between two positive onstants, and jrg Rj is bounded, (b) The Ri i urvature of the metri g has a lower bound, ( ) the Paneitz urvature is absolutely integrable, i.e. Z M jQjdvg < 1: Then M = S 4 n fpi gki=1 . It is easy to produ e a large number of su h metri s. The basi example is the in nite ylinder (R  S 3 ; dt2 + d 2 ) whi h is onformally equivalent to (R4 n f0g; jdxj2). The ylinder metri has R = 6; jRi j2 = 12; Q = 0. Thus given a domain = S 4 n fpi gki=1 , oneSsimply glues the ylinder metri s on Br (pi ) n fpi g to the spheri al metri on S 4 n Br (pi ). To make the s alar urvature positive over the glueing regions Br (pi ) n Br (pi ), one has to take  suÆ iently small. We remark that given any simply onne ted, lo ally onformally at, omplete manifold M of dimension n  3, there always exists an immersion  : M ! S n su h that the lo ally onformally at stru ture of M is indu ed by . This immersion  is alled the developing map of M . Under the further assumption that the s alar urvature is positive, S hoen-Yau proved that the developing map is inje tive. Therefore, any su h4 manifold an be onsidered as a subdomain of S n with a omplete metri g =  n 2 g where g is the standard metri on the sphere S n . Combining this result with Theorem 2.1 above, we obtain: 13 Corollary 2.2. Suppose M is a simply onne ted, lo ally onformally at, omplete 4-manifold satisfying onditions (a), (b) and ( ) as in the statement of Theorem 1.1 above, then M is onformally equivalent to S 4 n fpi gki=1 . In the remainder of the se tion, we will prove Theorem 2.1. Suppose M is a subdomain of S n , for onvenien e, we hoose a point P in M and use stereographi proje tion whi h maps S n to Rn and P to in nity; then we may identify M as 4 M = ( ; u n 2 jdxj2 ), where  Rn and jdxj2 is the standard metri of Rn . In the following we will estimate the size of the onformal fa tor u(x) for x 2 in terms of the Eu lidean distan e d(x) = distan e(x;  ). First we have the following lower bound estimate from ( [SY, Theorem 2.12, Chapter VI℄): Lemma 2.3. Suppose M = ( ; g = u n 2 jdxj2 ), where (a) jRj and jrg Rj both bounded, (b) the Ri i urvature has a lower bound. Then there exists a onstant C > 0 su h that 4 u(x)  Cd(x) n 2 2 for all x 2  Rn ; and suppose : (2.1) We remark that in the statement of Theorem 2.12, Chapter VI in [SY℄, a stronger assumption that M has bounded urvature is listed for the above result. But it is lear from the proof (e.g. applying method of gradient estimate), that assumptions (a) and (b) are suÆ ient for the on lusion. In the ase when M = ( ; u2 jdxj2 ) where  R4 and satis es the assumptions (a) and ( ) in Theorem 1.1, we an also establish the upper bound of u in terms of the distan e fun tion d. Lemma 2.4. Suppose M = ( ; u2 jdxj2 ) is a omplete manifold su h that (a) its s alar urvature R satis es 0 < R0  R  R1 and jrRjg  C , where R0 ; RR1 ; C are onstants, and ( ) jQju4 dx < 1. Then there exists some onstant C so that u(x)  Cd(x) 1 for all x 2 : (2.2) Our proof of the lemma uses a blow-up argument whi h was used by S hoen to obtain the upper bound for onformal metri s with onstant s alar urvature. Thus what we have done here is to repla e the onstant s alar urvature ondition by the integral bound of the Paneitz urvature and the ondition (a). The proof we give below for Lemma 2.4 depends on the following simple result, whi h is a spe ial ase of Theorem 4.1 in [CQY℄. We present the proof here to make the paper self- ontained. Lemma 2.5. On (R4 ; u2 jdxj2 ), the only metri with Q  0 and R  0 at in nity is isometri to (R4 ; jdxj2). Proof of Lemma 2.5. Re all the transformation formula for the s alar urvature 14 
w + jrwj2 = Je2w (2.3) where J = R6 and R is the s alar urvature for the metri (R4 ; e2w jdxj2 ). Noti e that when Q  0, then (
)2 w = Qe4w = 0, thus
w is a harmoni fun tion , hen e Z 1
w(x0 ) = jBr (x0)j Br (x0 )
wd Z 1 2 = jBr (x0 )j Br (x0 ) (jrwj + J )d: Thus by our assumption that J is non-negative, we have by taking r ! 1, for ea h x0 2 R4 ,
w(x0 )  0: Applying the Liouville theorem for bounded harmoni fun tions to
w we on lude
w = 0 . It follows that any partial derivative of w is harmoni , i.e.
wxi = 0: Apply the mean value theorem again, we have jwxi (x0 )j2 Z Z 1 1 =j wxi d j2  jr wj2 d: jBr (x0 )j Br (x0 ) jBr (x0)j Br (x0 ) But from (2.3) above we have jrwj2 = C0 Je2w ; hen e we on lude similarly as before that for ea h x0 2 R4 , jwxi (x0)j2  C0 : Hen e all partial derivatives of w are onstants. It follows that
w  C0  0. Hen e all partial derivatives of w vanish. Thus w is a onstant. Proof of Lemma 2.4. Assume (2.2) does not hold, then there exists a sequen e of points xi in with d(xi ) ! 0, and u(xi )d(xi ) = Ai ! 1; as i ! 1: By passing to a subsequen e of xi , we may also assume that B (xi ; 21 d(xi )) are disjoint and it follows from our assumption ( ) on the integrability of the Paneitz urvature Q that Z jQju4dx ! 0; as i ! 1: (2.4) B(xi ; 12 d(xi )) 15 We will now re-s ale u at ea h point xi . To do so, at ea h point xi , denote i = 1 2 d(xi ) and de ne fi (y ) = (i d(y; xi))u(y ): Then fi (xi ) = i u(xi ) = 21 Ai , and fi (y ) = 0 for all y 2 B (xi ; i ). Thus there exists some point yi in B (xi ; i ) for whi h f (yi ) = maxff (x) : x 2 B (xi ; i )g: We let i = u(yi ); and set vi (x) = i 1 u(i 1 x + yi ): Denote ri = 21 (i d(xi ; yi )) and Ri = ri u(yi ); then x 2 B (0; Ri) if and only if the orresponding point y de ned as y = i 1 x + yi is in B (xi ; ri ). We laim the following properties (2.5) to (2.7) hold for vi : vi (0) = i 1 u(yi ) = 1; (2.5) 0 < vi (x)  2 for x 2 B (0; Ri); and Ri ! 1 as i ! 1; (2.6) and
vi = Ji vi3 ; with jrJi (x)j uniformly bounded for x 2 B (0; Ri); (2.7) where Ji (x) = J (i 1 x + yi ). To verify (2.6), rst we have 2Ri = (i d(xi ; yi ))u(yi )  i u(xi ) = 12 Ai ! 1 as i ! 1 by the maximality of the hoi e of yi and (2.3). Furthermore we have for y =  i 1 x + yi u(y ) u(y )(i d(y; xi)) vi (x) = = u(yi ) u(yi )(i d(y; xi )) i d(yi ; xi ))  u(yui )((y)( i i ri ) i   r  2 for x 2 B (0; Ri): i i The equation in (2.7) is a dire t re-s aling of the s alar urvature equation of the metri g = u2 jdxj2 . The gradient estimate of Ji follows from our assumption (a) on the gradient bound of R with respe t to g and (2.6); we may see this as:   jrx Ji (x)j = jrx (J (i 1x + yi ))j = ji 1ry J (y)j  C6 i 1 u(y)  C3 ; where y = i 1 x + yi . It follows from (2.7) that, by taking subsequen e if ne essary, we have Ji ! J1 in Clo (R4 ) 16 for some J1 2 C (R4 ) and J1  16 R0 > 0. Hen e some subsequen e of vi onverges uniformly on ompa t in C 1; (R 4 ). Thus it follows from (2.7) that by taking another subsequen e, we also have vi ! u1 in Clo2; (R4) (2.8) where u1 2 C 2; (R4 ), u1 (0) = 1, and
u1 = J1 u31 in R4 : (2.9) wi ! w1 in Clo2; (R4 ): (2.10) By the maximum prin iple we know that u1 (x) > 0 for all x 2 R4 , whi h implies that, if we let wi = log vi and w1 = log u1 , and passing to a subsequen e we have We now laim that
2 w1 = 0 in R4 : To see this, for any  2 C 1 (R4 ) with ompa t support, we have (2.11) Z Z j
w1
j = j ilim !1
wi
j = j lim i!1 Z  C ilim !1  C ilim !1  C ilim !1 = 0:
2 wi j Z ZK j
2 wi j where K is the support of  K Z i Ki j
2wj where Ki = fy : y yi 2 i 1 K g (2.12) 2jQje4w for any  2 C 1 (R4 ) with ompa t support. The last step in above argument follows from (2.4), whi h is a onsequen e of our assumption ( ) in the Lemma. Hen e we have u1 ; w1 2 C 1 (R4 ), and the metri u1 2 jdxj2 is a metri on R4 with Q  0 and R  0. It then follows from Lemma 2.5 that u1 is a onstant fun tion, in ontradi tion with equation (2.9). We have thus nished the proof of Lemma 2.4. We will now estimate the size of the integral of Q over suitable subset of in terms of the integral of the boundary urvature T (as de ned in the introdu tion) via the Chern-Gauss-Bonnet formula. It will be advantegeous to onsider domains formed by level sets of the onformal fa tor u = ew . We will derive a formula for the boundary integral. We onsider M = ( ; u2 jdxj2 ) where  R4 , and denote the level set for the onformal fa tor u = ew by U = fx : 1  u  g; and S = fx : u = g: Also, n denotes the normal derivative ( hosen so that w n 17  0) . (2.13) Lemma 2.6. Suppose that M = ( ; u2 g0 ) is a omplete Riemannian manifold. Then on the level set S where  is a regular value for u, we have: Z d n
wd = d S + Z S Z S (n w)3 d + (n w)3 d + Z S Z J2 S Jn e4w n w Z we2w d + 2 U J jruj2 dx  d: (2.14) Proof. Observe that in terms of w = log u, the s alar urvature equation be omes
w + jrwj2 = Je2w as in (2.3). Thus when restri ted to the set S , we have n n w + Hn w + (n w)2 = Je2w ; (2.15) where H is the mean urvature onR S . d To al ulate the derivative d S fd we use the rst variation formula n d = 1 d Hd , and the hain rule d = ew n w n . Thus Z Z Z df f d fd = d + Hd: w d S S d S e n w In parti ular we have Z Z Z Z d ( w)3 ew d =3 n wn n wd + (n w)3 d + (n w)2 Hd d S n S S S Z  Z  =2 n wn n wd J (n w)e2w d: S S (2.16) On the other hand, Z S n
wd = Z =2 SZ n ((n w)2 + Je2w )d S n wn n wd + Z S Combine (2.16) and (2.17), we obtain Z Z Z (n J )e2w d + 2 Z J (n w)e2w d: S (2.17) Z d (n w)3 ew d + 3 J (n w)e2w d + (n J )e2w d: n
wd = d S S S S (2.18) 18 Next we ompute Z Z Z   w d J (n w)e3w d = (n J )e2w d + J n n e2w d d S S SZ n w Z +3 J (n w)e2w d + Je2w Hd = Thus S Z Z S n S Z Je2w d S Z J2 Z e4w d + 2 J (n w)e2w d: n w S (2.19) Z d d n
wd = (n w)3 ew d + J (n w)e3w d d d S S S Z Z e4w d: + J (n w)e2w d + J2 S S  n w (2.20) Or, equivalently, sin e ew = , Z d n
wd =  d S + Z S Z (n S ( n w)3 d + w)3 d + 2 Z S Z S J (n J ( n w)e2w d w)e2w d +  Z S J2 e4w d: n w (2.21) Finally we noti e that by the o-area formula, we have Z S J ( n w)e2w d Z Z d = J jrujd =  J jruj2 dx: d S U (2.22) Substituting (2.22) to (2.21), we obtain (2.14) and nish the proof of the lemma. We now state a simple overing lemma whi h we will use later in the proof of Theorem 2.1. Lemma 2.7. Suppose that  is a ompa t subset of R 4 . Then 8 < Ns3 ; for any N > 0 if dim() = 0 and H 0 () = 1 jfx : dist(x; ) = sgj  : 3 3 if dim() = > 0: Cs ; for = 4 (2.23) Where dim() denotes the Hausdor dimension of the set , and H denotes the Hausdor measure of the set. Proof. By a standard overing lemma, we have K = K (r) balls B (zi ; r) overing for  su h that B (zi ; 15 r) \ B (zj ; 15 r) = ; for i 6= j . And by the de nition of Hausdor measure we also know CKr  H (): 19 Now noti e that 1 B (zi ; r)  fx : dist(x; )  rg: 5 i [ we therefore have jfx : dist(x; )  rgj  CKr4  CH ()r4 ; whi h, by the isoperimetri inequality, implies jfx : dist(x; ) = rgj  C (H ()) 43 r3 3 4 : This is easily seen to imply (2.23). So we have proved the lemma. Proof of Theorem 2.1. We identify (M; g ) as ( ; u2 jdxj2 ) for some subset of R4 as before. Noti e that all ends of M is in bounded region inside . Apply integration by parts, we get Z U 2Qe4w dx = Z Z
2 wdx = U S1 Z n
wd + S n
wd: (2.24) Apply formula (2.15) in Lemma 2.4, we obtain Z U Z  = + Z S S1 S ( n (n w)3 d + w)3 d + Z S d  d V ( ); where V () is de ned as: V ( ) = n
wd n
wd SZ d = d Z 2Qe4w dx Z S (n Z S J2 w)3 d + J (n w)e2w d + 2 e4w d n w Z S J (n w)e2w d + 2 We re all the s alar urvature equation
u = Ju3 in : Thus Z U J jruj2 dx  J Z 0  J0 (  J02 U Z U Z  U jruj2 dx Ju4 dx Z S1 u4 dx; 20 Z U Z U un ud + J jruj2 dx
(2.25) J jruj2 dx: (2.26) un ud ) (2.27) Z S where J  J0 > 0 as assumed in (a). Consequently, V ()  2J02 Z U u4 dx: (2.28) To estimate the growth of V we use the lower and upper estimates of the onformal fa tor u as in Lemma 2.3 and Lemma 2.4; thus we may repla e the region U by D = fx : C1  d(x;  )  C2  1g  U : (2.29) Therefore we have V ( )  Z D u4 = Z C1 Z fx:d(x; C2  )=sg u4 dds by the o-area formula. Hen e V ( )  C Z C1 C2  jfx : d(x;  ) = sgjs 4ds: (2.30) We now estimate the size of the set  by Lemma 2.7. In the ase dim( )= positive, we have from (2.23) and (2.30) that V ( )  C ((  ) C2 1 ); C1 is (2.31) for = 43 whi h is positive. In the ase when dim( ) is zero, we have either the zero Hausdor measure of the set (i.e. number of points in the set) is nite; then we have proved the theorem; or we have jfx : d(x;  ) = sgj  Ns3 (2.32) for any number N > 0. Hen e V ( )  N Z C1 C2  1 ds = N log  C: s (2.33) In either ase of (2.31) or (2.33), we on lude that there exists at least a sequen e of i ! 1 as i ! 1 su h that i are all regular values (due to Sard's theorem) and d  i V ( i )  N (2.34) d for any number N > 0. But in view of the equality (2.25), this ontradi ts with our assumption ( ) that Q is integrable. We have thus nished the proof of the theorem. 21 Corollary 2.7. If (  S 4 ; e2w g0 ) is a omplete onformal metri satisfying (a) (b) and ( ) and in addition Q is a onstant, then: either 1. (  S 4 ; e2w g0 ) = (S 4 ; g0 ); or j2 2. (  S 4 ; e2w g0 ) = (R 4 n f0g; jjdx xj2 ). Proof of Corollary 2.3: A ording to Theorems 1 and 2, = S 4 nfp1 ; :::; pk g; and if k  1 the inequality (2.1) shows that the metri has in nite volume. Thus in ase k = 0, the standard metri on S 4 is the unique solution of the equation Q = 3 up to onformal transformations. In ase k  1 we must have Q = 0. Hen e the euler number an only have two possibilities:  = 1 or  = 0. In the former we have is onformally R4 , then the metri must be standard a ording to Lemma 2.5, but the s alar urvature is zero hen e this ase annot o ur. The remaining possibility is the ase  = 0 so that is onformally R4 n f0g, and Q = 0. To determine the metri e2w jdxj2 , we use Proposition 1.12 to on lude that w is of the form w(x) = log jxj + h(x); for jxj > 1 where h is a biharmoni fun tion whi h extends over in nity and  1 due to ompleteness of the metri . On the other hand in the pun tured ball we have w(x) = log jxj + k(x); for jxj < 1 where k is a biharmoni fun tion on jxj < 1 and  1 due to the ompleteness requirement. On the other hand the generalized Chern-Gauss-Bonnet formula (0.7) requires +  0. Hen e 2= 1; = 1 and we on lude that h = k = onstant, j and therefore e2w jdxj2 = jjdx xj2 up to a homothety as laimed. x3. Non-simply onne ted manifolds In this se tion we will prove Theorem 3.1. The main idea is to lo alize the argument in the proof of Theorem 2.1 in the previous se tion and appeal to some well known elementary fa ts in the theory of Kleinian groups. Suppose (M; g ) is a lo ally onformally at manifold with positive s alar urvature, then by the result of S hoen-Yau, the universal over M~ an be embedded as a domain in the 4sphere. Hen e the fundamental group a ts on S 4 as a dis rete group of onformal transformations with a domain of dis ontinuity ( ) whi h ontains M~ . (Here as in the rest of this se tion, we refer to [Ra℄ for standard notations and de nitions of Kleinian groups.) The limit set L onsists of a umulation points of orbits of . The dis rete group also a ts as hyperboli isometri s on the interior B of S 4 . We re all the following de nitions of limit points. A point p 2 S 4 is alled a oni al limit point of the group if there is a point x 2 B , a sequen e fgi g  , a hyperboli ray in B ending at p and a positive number r su h that fgi (x)g onverges to p within hyperboli distan e r from the geodesi . A point p is a usped limit point of a dis rete group if it is a xed point of a paraboli element of that has a usped region. To explain the notion of a usped region, we identify S 4 as R4 and onjugate the point p to in nity in the upper half spa e R 5+ in R 5 , 22 and onsider the stabilizer 1 of 1. 1 is a dis rete subgroup of isometri s of R 4 of rank 0 < m  4. Let E be the maximal 1 -invariant subspa e su h that E= 1 is ompa t. Denote by N a neighborhood of E in R 5+ and set U = R 5+ n N . Then U is an open 1 invariant subset of R 5+ . The set U is said to be a usped region for based at 1 if and only if for all g in n 1 , we have U \ gU = ;. De nition. A onvex polyhedron P in the hyperboli spa e is alled geometri ally nite if and only if for ea h point x 2 P \ S 4 there is an open Eu lidean neighborhood of x that only meets the fa es of P in ident with x. A dis rete group of onformal transformations of S 4 is alled geometri ally nite if and only if has a geometri ally nite onvex fundamental polyhedra. An important aspe t of geometri ally nite Kleinian groups is that its limit set onsists only of oni al limit points and usped limit points ([Ra℄, Theorem 12.3.5, p.512) Now we are ready to state and prove our main theorem in this se tion. Theorem 3.1. Suppose M is a lo ally onformally at omplete 4-manifold whi h satis es: (a) The s alar urvature is bounded between two positive onstants, and jrg Rj is bounded, (b) The Ri i urvature has a lower bound , ( ) The Paneitz urvature is absolutely integrable, i.e. Z M jQjdvg < 1; (d) The fundamental group of M a ting as de k transformation group is a geometri ally nite Kleinian group without torsion. Then M = M n fpi gki=1 where M is ompa t manifold with a lo ally onformally at stru ture. Remarks 1. It is worthwhile to point out that M is sometimes alled the onformal ompa ti ation of M . 2. Suppose M satis es all assumptions in Theorem 3.1 ex ept its fundamental group being torsion free. Instead, its fundamental group has a nite index subgroup without torsion. Then, by passing to a nite overing M 0 of M , we may apply Theorem 3.1 to M 0 . For example, we have Selberg's lemma ([Ra, p. 327℄) whi h states that any nitely generated dis rete matrix group ontains a nite index subgroup without torsion. So, in the ases when the fundamental group of M is nitely generated then Theorem 3.1 applies to the torsion free subgroup, this means that the ends are onformally the quotient of pun tured 4-ball. Proof of Theorem 3.1. Denote  :M~ ,! S 4 :# M 23 where  is the developing map from the universal overing M~ of M into S 4 , whi h is an embedding. The holonomy representation of the fundamental group of M then be omes a Kleinian group in the onformal transformation group of S 4 , whi h is assumed to be geometri ally nite and torsion free. Let = (M~ ). Clearly any point in is a ordinary point for , that is  ( ): the domain of dis ontinuity of . We are interested in the set  = S 4 n . If M is ompa t, then  = L( ) is the set of all limit points of and = ( ), therefore M = ( )= . But, in our ase, we have [  = ( \ ( )) L( ); and L( ) onsists of only oni al limit points and usped limit points. We laim: Claim 1. Every point in  \ ( ) is an isolated point in S 4 , and Claim 2. There is no usped limit points ex ept possibly usped limit points of rank four in whi h ase the losure of the fundamental region for the usp does not meet the limit set. First we apply the proof of Theorem 2.1 in previous se tion to prove Claim 1. A ording to Theorem 2.9 and Theorem 2.11 in Chapter VI of [SY℄, dim( )  d(M ) < 1, hen e is a totally dis onne ted set ( f Lemma 4.1, Chapter 4 in [Fa℄ ). Therefore, for any x 2  \ ( ), thereTexists a ball B (r; x) su h that B (r; x) T B (r; x) = ; for all 2 and  B (r; x)  = ;. Sin e the Q urvature is absolutely integrable over \ B (r; x), we an restri t the onformal metri to B (r; x) \ and apply the argument in the proof of Theorem 2.1 to on lude that  \ B (r; x) onsists of at most nite number of points in luding x, thus in parti ular x is an isolated boundary point. To prove Claim 2, we re all, from the de nition of the usped limit points, a usped limit point p is a xed point of a paraboli element p in and there is a usped region U for based at p. The usped region based at p restri ted to the 4-sphere gives a onformal oordinates hart for M at the end Ep around p of the following form ( f: Chapter 12 in [Ra℄): (M; g )  (Ep ; g ) = (Sm ; 2 gm ) for some Sm ; 1  m  4, where S1 = T 1  fx 2 R3 : jxj  K g, S2 = T 2  fx 2 R2 : jxj  K g, S3 = T 3  fx 2 R : jxj  K g, S4 = T 4 , where T k is a at torus of dimension k, K > 0 is a large positive number, gm is the produ t metri on ea h Sm , and  is a positive smooth fun tion. Now we will show that su h ends Ep an not exist in our ase. Lemma 3.2. There is no omplete onformal metri 2 gm on (Sm ) su h that its s alar urvature bounded from below by a positive number, for m = 1; 2; 3. Proof. Case (i): When m = 1. Re all the s alar urvature equation on S1 : 1 2 (r r  + r  + 2
S 2  +
T 1 ) = J3 r r 24 for r = jxj , r 2 [K; 1) the norm of a point x in R3 . We take the average of  over S 2  T 1 for ea h r and get 2 (r r  + r )  J0 ()3 r where J  J0 as assumed. Now take a hange of variables ( t e therefore =r ; = r tt + t  J0 (3.1) 3 (3.2) We will show that attains zero or in nity at some nite t, whi h will be a ontradi tion. First we observed that, if t (t0 )  0 at ertain t0 , then tt (t0 ) < 0. Therefore we have t  < 0 and tt  0 for all t  t1 for some t1 > t0 , whi h implies has to be zero at some nite t. Thus, we may assume t > 0 for all t. Next we observed that, if t J0 3  0 at some t0 , then tt (t0 )  0 and therefore < 0 for all t  t2 for some t2 > t0 ; then t an not be positive for all tt (t)  t. Thus, we may assume 3 t J0 > 0 for all t. But this implies d 1 ( ) < 2J0 ; dt 2 whi h is impossible unless goes to in nity at some nite t. This nished the proof of ase (i). Case (ii): When m = 2, then the s alar urvature equation for S2 is: 1 1 (r r  + r  + 2
S 1  +
T 2 ) = J3 r r for r 2 [K; 1) the norm of a point in R2 . Then we take the average over S 1  T 2 and the hange of variables as in (3.1) to get tt + 2 t  J0 3 + : (3.3) A similar argument as in the above proves the lemma for S2 . Case (iii): For m = 3, then the s alar urvature equation on S3 be omes: (r r  +
T 3 ) = J3 for r get 2 [K; 1). Then again we take the average and the hange of variables, and tt + 3 t  J0 3 + 2 : 25 (3.4) On e again, a similar argument as in the previous ases establishes the lemma for S3 . Case (iv): When m = 4, in this ase S4 = T 4 is ompa t, hen e its fundamental domain in the domain of dis ontinuity an be hosen to have its losure bounded away from the limit set. We have thus nished the proof of the lemma. To ontinue the proof of Theorem 3.1, we apply Lemma 3.2 to on lude that the limit set onsists of either oni al limit points or usp of rank four. Sin e xed points of either hyperboli or paraboli elements in are all in the limit set L( ) of , a ts on ( ) has no xed points. Thus ( )= = M is a manifold with a lo ally onformally at stru ture. We onsider a fundamental domain F whi h satis es: ( )= [ 2 F and F \ F = ;; 8 2 : And let C = F the losure of F in S 4 . Sin e all points in L( ) are either oni al or usps of rank four, the losure of F does not meet the limit set, hen e C  ( ). This proves that M = C= is ompa t sin e C is ompa t. In the mean while, we have M = (C n ( \ ( )). Now sin e  \ ( ) are all isolated points without a umulation points in ( ), C \ ( \ ( )) must be a nite point set. Therefore M = M n fpi gki=1 for some k < 1. We have thus nished the proof of the theorem. Referen es [ABR℄ S.Axler, P. Bourdon and W. Ramey; \Harmoni Fun tion Theory", GTM 137, Springer-Verlag 1992. [BCY℄ T. Branson, S-Y. A. Chang and P. C. Yang; \Estimates and extremal problems for the log-determinant on 4-manifolds", Communi ation Math. Physi s, Vol. 149, No. 2, (1992), pp 241-262. [CQ℄ S.-Y. A. Chang and J. Qing; \The zeta fun tional determinants on four manifolds with boundary I - the formula", J. Fun t. Anal. Vol. 147, No. 2, (1997), pp 327-362. [CQY℄ S.-Y. A. Chang, P. C. Yang and J. Qing; \On the Chern- Gauss-Bonnet Integral for onformal metri s on R 4 ", preprint, 1998, to appear in Duke Math. Jour.. [CY℄ S-Y.A. Chang and P. C. Yang, "Extremal metri s of zeta fun tion determinants on 4-manifolds ", Annals of Math. 142 (1995), pp 171-212. 26 [CG℄ J. Cheeger and M. Gromov; \On the hara teristi numbers of omplete manifolds of bounded urvature and nite volume", Di erential geometry and omplex analysis, 115{154, Springer, Berlin-New York, 1985. [CV℄ S. Cohn-Vossen; \Kurzest Wege und Totalkrummung auf Fla hen", Compositio Math. 2 (1935), pp 69-133. [Fa℄ K. J. Fal oner; \The goemetry of fra tal sets", Cambridge University Press, 1985. [F℄ R. Finn; \On a lass of onformal metri s, with appli ation to di erential geometry in the large", Comm. Math. Helv. 40 (1965), pp 1-30. [GW℄ R. Greene and H. Wu; C 1 onvex fun tions and manifolds of positive urvature. A ta Math. 137 (1976), no. 3-4, 209{245. [Hu℄ A. Huber; \On subharmoni fun tions and di erential geometry in the large", Comm. Math. Helv. 32 (1957), pp 13-72. [MP℄ R. Mazzeo and F. Pa ard; Constant s alar urvature metri s with isolated singularities, Duke Math. Jour. 99 (1999) 367-418. [P℄ S. Paneitz; \A quarti onformally ovariant di erential operator for arbitrary pseudo-Riemannian manifolds", Preprint, 1983. [Po℄ D. Polla k; Compa tness results for omplete metri s of onstant positive s alar urvature on subdomains of S n ; Indiana Math. Jour. 42 (1993), pp 1441-1456. [Ra℄ J. Rat li ; \Foundations of hyperboli manifolds", 1994 Springer Verlag. [S℄ R. S hoen; \The existen e of weak solutions with pres ribed singular behaviour for a onformally s alar equation", Comm. Pure and Appl.Math. XLI (1988), pp 317-392. [SY℄ R.S hoen and S.T. Yau; \Le tures on Di erential Geometry", International Press, 1994. Sun-Yung Ali e Chang, Department of Mathemati s, Prin eton University, Prin eton, NJ 08544 & Department of Mathemati s, UCLA, Los Angeles, CA 90095. E-mail address : hangmath.prin eton.edu Jie Qing, Department of Mathemati s, University of California, Santa Cruz, Santa Cruz, CA 95064. E-mail address : qingmath.u s .edu Paul Yang, Department of Mathemati s, University of Southern California, Los Angeles, CA 90089. E-mail address : pyangmath.us .edu 27