ON THE CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS ON R 4

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1 Vol. 03, No. 3 DUKE MATHEMATICAL JOURNAL 2000 ON THE CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS ON SUN-YUNG A. CHANG, JIE QING, PAUL C. YANG. Introduction. An important lmark in the theory of surfaces is the introduction of the notion of complete open surface by Hopf Rinow [HR]. Subsequently, Cohn-Vossen [CV] studied the Gauss-Bonnet integral for such a surface M with analytic metrics. Cohn-Vossen also showed that if the Gaussian curvature K is absolutely integrable, then.) Kdv M 2πχM), M where χm) is the Euler number of M. Later, Huber [Hu] extended this inequality to metrics with much weaker regularity. More importantly, he proved that such a surface M is conformally equivalent to a closed surface with finitely many punctures. The deficit in formula.) has an interpretation as an isoperimetric constant. On a complete open surface with Gaussian curvature absolutely integrable, one may represent each end conformally as R 2 \K for some compact set K. We consider the isoperimetric ratio L 2 r) ν = lim r 4πAr), where Lr) is the length of the boundary circle ={ x =r}, Ar) is the area of the annular region Br)\K. For a fairly large class of complete surfaces, Finn [F] showed that χm).2) Kdv M = ν j, 2π M where the sum is taken over each end of M. For more recent development on the subject of complete surface of finite total curvature the reader is referred to the work of Li Tam [LT]. Except for the work of Cheeger Gromov [CG] on the Chern-Gauss-Bonnet formula for manifolds with bounded geometry, there is very little known about the situation in higher dimensions. Received 0 February Mathematics Subject Classification. Primary 53A30, 35J35; Secondary 53A55, 25J6. Chang s work partially supported by National Science Foundation grant number DMS Qing s work partially supported by National Science Foundation grant numbers DMS DMS Yang s work partially supported by National Science Foundation grant number DMS

2 524 CHANG, QING, AND YANG In this paper we consider a generalization of.2) in dimension 4. For conformal geometry in dimension 4, the Paneitz operator ) 2 P = 2 +δ RI 2Ric d 3 where δ denotes the divergence, d is the differential, R is the scalar curvature, Ric is the Ricci tensor) plays the same role as the Laplacian in dimension 2 cf. [P], [BCY], [CQ], for example). The Paneitz operator enjoys the following invariance property under conformal change of metric g = e 2w g 0 : the Paneitz operator transforms by P g = e 4w P g0. For the conformal metric g = e 2w g 0, the Paneitz operator applied to the conformal factor w calculates a fourth-order curvature invariant Q:.3) P g0 w +2Q g0 = 2Q g e 4w, where Q = { R + 4 }.4) 2 R2 3 E 2 E is the traceless Ricci tensor. The Q curvature invariant is related to the Chern- Gauss-Bonnet integral in dimension 4; we have χm)= W 2 ).5) 4π 2 M 8 +Q dv, where W is the Weyl tensor M is a compact, closed 4-manifold. More generally, when the manifold has a boundary, Chang Qing [CQ] have defined a boundary operator P 3 its associated boundary curvature invariant T, as follows:.6) P 3 w +T g0 = T g e 3w. Then the Chern-Gauss-Bonnet integral is supplemented by χm)= W 2 ) 4π 2 8 +Q dv +.7) 4π 2 L+T)d, M where Ld is a pointwise conformal invariant. Our motivations are twofold. First, we are searching for the analogue of the previously mentioned result of Finn. Second, we would like to find a geometric interpretation of the fourth-order curvature invariant Q which we call Paneitz curvature, according to the relation in equation.3)). We are fortunate to find both in one formula. Here we take the initial analytic step study complete conformal metrics e 2w dx 2 on. For interesting applications in global geometry it would be desirable to establish some criteria for conformal compactification, at least for locally conformally flat 4-manifolds. We will continue to study this question in a forthcoming article. M

3 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 525 We now state our results. By solving an ODE, we first verify the following theorem. Theorem.. Suppose that w is a radial function on e 2w dx 2 is a complete metric with Q e 4w dx <. And suppose that its scalar curvature is nonnegative at infinity. Then.8) Qe 4w dx 4π 2 χ ) = 4π 2, vol 4π 2 Qe 4w Br 0) )) 4/3.9) dx = lim r 4 2π 2) /3 vol Br 0) ). Remark.2. ) By nonnegativity of scalar curvature at infinity we mean that the scalar curvature Rx) is nonnegative when x is sufficiently large. It is easy to see that without the additional condition on the scalar curvature Theorem. does not hold. For instance, suppose that e 2v dx 2 is the stard metric for the 4-sphere S 4. Then the metric e 2v+r2) dx 2 is a complete metric on ;but 4π 2 Q v+r 2e 4v+r2) dx = 4π 2 2 v +r 2) dx = 4π 2 2 v Q S 4e 4v dx = 3 S4 4π 2 = 2 >. = 4π 2 2) For domains in R n for n>2, it turns out that there is more than one isoperimetric constant to be considered. These isoperimetric constants can be defined as the ratio of pairs of mixed volume see [BZ], [T]) of the domain. It turns out that in our setting, after suitable normalization, all these isoperimetric ratios tend to the same limit as in.9). We have established a generalized form of Theorem. also Theorems.3.4), with respect to all these isoperimetric ratios. In his attempt to prove.2) for any complete open surface with absolutely integrable Gaussian curvature, Finn [F] has found an interesting intermediate class of metrics for which.2) holds. That is the class of normal metrics, which are the conformal metric e 2w dx 2, where w is a constant plus a potential see Definition 3.). We found the same phenomenon in 4-dimension. Theorem.3. Suppose that the metric e 2w dx 2 is a complete normal metric on ; then both.8).9) hold. The notion of normal metrics arises naturally. In the recent work of several authors see [CLi], [CL], [CY], [L], [Xu]), they have all pointed out that it is essential to verify that a metric is normal in order to control the asymptotic behavior for metrics at infinity. Fortunately, we are able to show that nonnegativity of the scalar curvature at infinity) ensures that the metric is normal.

4 526 CHANG, QING, AND YANG Theorem.4. Suppose that the Paneitz curvature of a metric e 2w dx 2 on is absolutely integrable suppose that its scalar curvature is nonnegative at infinity; then the metric is normal. We believe that one also can verify a metric to be normal if one assumes some decay conditions of the Paneitz curvature such as those given in [CL]. Meanwhile, Theorem.4 sheds some light on estimating asymptotic behavior for a metric e 2w dx 2 on under appropriate geometric conditions. Finally, combining Theorems.3.4 we have our main theorem. Main theorem. Suppose that e 2w dx 2 on is a complete metric with its Paneitz curvature Q absolutely integrable, suppose that its scalar curvature is nonnegative at infinity. Then both.8).9) hold. The paper is organized in the following manner. In Section 2, we will study the radially symmetric case. Our approach is to first find a particular good solution to the ODE, then to control the asymptotic behavior of the given metric. In Section 3, we prove Theorem.3. Our strategy is to reduce the problem to a radially symmetric one, then to apply the result in Section 2. In Section 4, we prove Theorem.4 hence our main theorem. Acknowledgments. Part of this work was done while the second author was visiting UCLA Princeton University while the third author was visiting Princeton University. They would like to thank both institutions for their hospitality. 2. Symmetric case. Consider a conformal metric,e 2w dx 2 ). We assume that the Paneitz curvature Q of the metric 2.) 2 w = 2Qe 4w on satisfies 2.2) Q e 4w dx <. For convenience, we often use cylindrical coordinates x =r = e t to rewrite equation 2.) as 2.3) 2 t 2 2 ) 2 t t 2 +2 ) w +t) = 2Qe 4w+t) ; t that is, 2.4) 2 t 2 2 ) 2 t t 2 +2 ) v = 2Qe 4v, t <t<, when we denote w +t by v.

5 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 527 In contrast to the situation on surfaces, where there is one isoperimetric ratio to consider, there exists a family of isoperimetric inequalities for mixed volumes, when the dimension of the manifold is greater than 2 cf. [BZ], [T]). In the case when is a convex domain in R n, following Trudinger [T] we define, for each m<n, the mixed volume V m ) = n n ) H n m [ ]dh n, m where H l [ ] is the lth symmetric form of the principle curvatures of where H n is the n dimensional Hausdorff measure. For m = n, define V n ) = volume of. Denote ζ m ) = Vm /m ); then the isoperimetric inequalities for mixed volumes 2.5) ζ m ζ l hold for all l m n. Motivated by these inequalities, we may similarly define, in our case when n = 4), 2.6) V 4 = vol{s t}) = S 3 t e 4v ds, V 3 = 2.7) e 3v ds, 4 s=t V 2 = 2.8) H e 3v ds = 2 s=t 4 S3 v t)e 2vt), V = H 2 e 3v ds = H 2 2 s=t 24 tr L 2) e 3v ds = s=t 4 S3 v t) ) 2 2.9) e vt). We also define the following isoperimetric ratios: C 3,4 t) = V 4/3 3 t) /2)π 2 ) /3 V4 t), 2.0) 2.) C 2,3 t) = C,2 t) = V 2 t) /2)π 2 ) /3 V 2/3 3 t), V 2/3 t) /2)π 2 ) /3 V /3 2 t), C 2,4 t) = C /3 3,4 t)c2/3 2,3 t), C,3 t) = C /4 2,3 t)c3/4,2 t), C,4 t) = C /9 3,4 t)c2/9 2,3 t)c2/3,2 t).

6 528 CHANG, QING, AND YANG Then we have 2.2) C 2,3 t) = C,2 t) = C,3 t) = v t). Furthermore, when both V 4 t) V 3 t) tend to infinity as t tends to infinity, we also have, via L Hôpital s rule, 2.3) lim C 3,4t) = lim C,4t) = lim C 2,4t) = lim t t t t v t). In the proof of Theorem 2., we have also established 2.3) for the general cases. On the other h, by the Chern-Gauss-Bonnet formula, we have 2.4) 4π 2 Qe 4v = Te 3v. s t As defined in [CQ, Remark 3.], in the special case of with cylindrical coordinate we have s=t 2.5) Therefore, 2.6) Te 3v = P 3 v = 2 v +2v. 4π 2 Qe 4v = s t 2 S3 v t)+4v t) ). Now to relate the total Paneitz curvature with any of the isoperimetric constants C l,m as defined above, we would like to show that, under suitable conditions, 2.7) lim t v t) = lim t v t) = 0. Thus, we turn to study the behavior of v. For convenience, we denote 2Qe 4v by F. Equation 2.4) is equivalent to the following ODE: 2.8) f 4f = F, <t<, where F dt <. We first find a special solution to 2.8). Denote f = Ct)e 2t, then Ct) satisfies C t) 4C t) = Ft)e 2t, or equivalently C t)e 4t) = Ft)e 2t. Thus, we can solve for Ct) as follows: t Ct) = e 4x Fy)e 2y dy dx 2.9) x = 4 e4t Fx)e 2x dx 4 t t Fx)e 2x dx.

7 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 529 Therefore, 2.20) f t) = 4 e2t Fx)e 2x dx t 4 e 2t t Fx)e 2x dx, 2.2) f t) = t {e 2t Fx)e 2x dx e 2t Fx)e 2x dx 8 t t } +K K 2 )+ Fx)dx Fx)dx, where t 2.22) K = lim t e2t Fx)e 2x dx, t t K 2 = lim t e 2t Fx)e 2x dx. Thus, we obtain 2.23) vt) = c 0 +c t +c 2 e 2t +c 3 e 2t +ft), for some constants c 0,c,c 2,c 3. Lemma 2.. K = K 2 = 0. Proof. 2.24) We prove, for instance, K 2 = 0 first: e 2t t Fx)e 2x dx = e 2t { T Fx)e 2x dx+ e 2t T) Fx) dx+ t T T } Fx)e 2x dx Fx) dx. Thus, if we take T<t, say, take T = /2)t, let t tends to, we obtain K 2 = 0. Similarly we have { T } e 2t Fx)e 2x dx = e 2t Fx)e 2x dx+ Fx)e 2x dx 2.25) t Thus, K = 0 by a similar argument. t T Fx) dx+e 2t T) Fx) dx. T

8 530 CHANG, QING, AND YANG As a direct consequence of Lemma 2., we have Corollary ) 2.27) lim f t) = t 8 lim t f t) = 8 Lemma 2.3. c 2 = 0. Proof. From 2.20) Lemma 2., we have which is zero. Also we have Fx)dx, Fx)dx. lim t f t) = 4 K, 2.28) v t) = w rr r2 +w r r, with r = e t tending to zero as t tends to negative infinity. Thus, from 2.23) we conclude that c 2 = 0. Lemma 2.4. c = /8) Fx)dx. Proof. Since v t) ast, this lemma follows from 2.23) 2.27). We now try to eliminate the c 3 coefficient of the e 2t term in vt). We notice that without any further assumption, in addition to Q being absolutely integrable, this is not always possible. For example, we may consider vt) = e 2t +t. Then Q = 0, the metric e 2v t) dx 2 = e 2r2 dx 2 is complete on. Lemma 2.5. Suppose we assume that the scalar curvature is nonnegative at infinity; then c 3 = 0. Proof. This is a consequence of the transformation formula for the scalar curvature. Denote R the scalar curvature of the metric. Then, with the cylindrical coordinate, we have 2.29) v v ) 2 + = 6 Re2v. Thus, if R 0 at infinity, it follows from 2.23), 2.26), 2.27), 2.29), Lemma 2.4 that c 3 = 0. Remark2.6. It is even easier to see that if v t) = O) as t, then c 3 has to be zero, in light of 2.23) 2.26). We will use this observation in the proof of Theorem 3.8 in the next section.

9 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 53 Theorem 2.7. Suppose that w is a radial function on, e 2w dx 2 is a complete metric with Q e 4w dx < the scalar curvature nonnegative at infinity. Then lim t v t) = 2.30) 4π 2 Qe 4w dx 0. Moreover, we have lim C l,mt) = 2.3) t 4π 2 Qe 4w dx, for all l m 4, where C l,m are as defined in 2.0) 2.). Proof. Under the assumptions, we have by Lemmas that 2.32) vt) = c 0 +c t +ft). Thus, we have v t) = f t), which tends to zero as t tends to infinity. We also have lim t v t) = c + lim f t) = Fx)dx = t 4 4π 2 Qe 4w. From the completeness of e 2w dx 2, we then conclude that lim t v t) 0. Thus, if this limit is strictly positive, then both V 4 t) V 3 t) tend to infinity as t tends to infinity; 2.3) then follows from L Hôpital s rule. On the other h, if v t) tends to zero lim t V 4 t) is bounded, then lim t e 4vt) = 0, hence lim t e kvt) = 0 for k 3. We again have lim t C 3,4 t) = 0 = lim t v t). Thus, we establish 2.3). 3. Normal metrics. In this section we first define normal metrics as a generalization of the definition of normal metrics in 2-dimension given by Finn in [F]. Then we verify that, for a complete normal metric on, the generalized Chern-Gauss-Bonnet formula.9) holds. Suppose that e 2w dx 2 is a metric on with its Paneitz curvature Q absolutely integrable, that is, 3.) Q e 4w dx <. Definition 3.. A conformal metric e 2w g 0 satisfying 3.) is defined to be normal if wx) = y 3.2) 4π 2 log x y Qy)e4wy) dy +C. So what are the normal metrics on? It turns out that this is not an easy question. But we show that a fairly large class of metrics on are normal in the following

10 532 CHANG, QING, AND YANG section. More precisely, we prove that, if the scalar curvature of the metric e 2w dx 2 is nonnegative at infinity, then the metric e 2w dx 2 is normal. As in the previous section, one defines the following mixed volumes on the manifold,e 2w dx 2 ): 3.3) 3.4) 3.5) 3.6) V 2 r) = 2 V r) = 2 V 4 r) = e 4w dx, B r V 3 r) = e 3w dσ x), 4 H e 3w dσ x) = 4 H 2 e 3u dσ x) = 4 r + w r ) e 2w dσ x), r + w ) 2 e w dσ x). r We also define C l,m r) as in the previous section. In the following, we will adopt some techniques used in [F] to compare V m r) as previously defined) with V m r), the mixed volume for the metric e 2 w dx 2 which may be considered to be the average of the metric e 2w dx 2 ), where w is defined as wr) = wx)dσ x). Lemma 3.2. Suppose that the metric e 2w dx 2 on is a normal metric. Then, for any number k>0, 3.7) e kw dσ x) = e k wr) e o), where o) 0 as r. Proof. Suppose e 2w dx 2 is a normal metric. We rewrite w as 3.8) wx) = 4π 2 B x /2 0) + 4π 2 = w x)+w 2 x). y log x y Qy)e4wy) dy +C \B x /2 0) y log x y Qy)e4wy) dy

11 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 533 Then we write 3.9) w x) = 4π 2 y /2) x + 4π 2 = f x )+w 0 x), y /2) x log y x Qy)e4wy) dy +C x log x y Qy)e4wy) dy where 3.0) { w 0 x) C + y η x C log η +C η x y /2) x η x y /2) x } x log x y Qy) e4wy) dy Qy) e 4wy) dy = o). Since 3.) Qy) e 4wy) dx <, log x x y log3, we may take η 0 while η x,as x ), then w 0 x) =o) as x. We then have 3.2) w x) = f x )+o), for some function f, which in turn implies 3.3) k wy) w2 y) ) dσ y) 0) 0) = log e kwy) w2y)) dσ y)+o). 0) y /2)r 0) Next, we study the term w 2 x)dσ x) 0) 0) 3.4) = { 8π 2 0) 0) y log x y } dσ x) Qy)e 4wy) dy.

12 534 CHANG, QING, AND YANG We also claim that the term 3.5) L = 0) 0) y log dσ x) x y 0) 0) is bounded for y /2)r. To prove the claim we first rewrite L as L 0) log y 0)\{x: x y θ y } x y dσ x) 3.6) + 0) 0) log y {x: x y θ y } x y dσ x) = L +L 2, log y x y dσ x) where θ is chosen to be smaller than /2 for example). Then it is easily seen that 3.7) 3.8) L log θ L 2 Br/ y )0) B log r/ y )0) {x: x y/ y ) θ} x y/ y ) dσ x), which is bounded because 3.9) θ r y 2, log / x y/ y ) is certainly integrable. This proves the claim 3.5). Thus, from 3.4) we conclude that 3.20) w 2 x)dσ x) = o) as r, 0) 0) when Q is absolutely integrable on,e 2w dx 2 ). Finally, we consider the term e w2x) 3.2) )dσx)= e w 2rσ ) )dσ. 0) B 0) 0) B 0) Following [F], we will estimate the term E M ={σ S 3 : w 2 rσ ) >M}. Wehave 3.22) M E M w 2 dσ { E M 8π 2 \B r/2 0) = 8π 2 Hy) Qy) e 4wy) dy. y r/2 E M log y rσ y } Qy) e dσ 4wy) dy

13 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 535 We will estimate the function Hy) for y r/2. Again we decompose H as Hy)= log y E M \{σ : rσ y /3) y } rσ y dσ 3.23) + log y E M {σ : rσ y /3) y } rσ y dσ = H +H 2. Clearly, 3.24) H log3 E M. It is more difficult to estimate the term H 2. To do this, we observe that for rσ y /3) y, wehave log y rσ y log y r + log σ y r log log σ y 3.25) r. Thus, we may estimate H 2, when E M is a 3-dimensional disk centered at y/r, perpendicular to y, of the size E M,as ) H 2 C E M +C E M log E M C 3.26) +log E M. E M Combining 3.22), 3.23), 3.24), 3.26), we obtain ) 3.27) M o) +log, E M where o) 0asr ; this then implies 3.28) Thus, 3.29) 0) = 0) + k B 0) E M Ce M/o). e kw 2 x) ) dσ x) e km ) E M dm = o), as r. Combining 3.3), 3.20), 3.29), we arrive at k wx)dσ x) = log 0) 0) 0) 0) e kwx) dσ x)+o),

14 536 CHANG, QING, AND YANG or equivalently, e kwx) dσ x) = e o) 0) e k wr). 0) Thus, we have established 3.7). As a direct consequence we have the following corollary. Corollary 3.3. Suppose that the metric e 2w dx 2 on is a normal metric. Then 3.30) V 3 r) = V 3 r) +o) ), 3.3) ) d d +o) dr V 4r) = dr V ) 4 r). To deal with V,V 2, we will first establish the following technical lemma. Lemma ) Suppose that e 2w dx 2 is a normal metric on. Then ) w k ) dσ x) = O r r k, for k =,2,3, 3.33) Proof. We observe that ) w 2 dσ x) = r w r r) = ) w 2 ) r)+o r r 2. w r dσ x). To prove 3.32), we have, for a normal metric e 2w dx 2, w 3.34) r x) = Kx,y)Fy)dy, where Kx,y) = /8π 2 x ) x 2 x y)/ x y 2 Fy) = 2Qy)e 4wy), which is integrable over. Then { ) w k dσ x) r Kx,y) k dσ x) Fy) dy } { } k Fy) dy.

15 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 537 Therefore, to prove 3.32) it suffices to prove ) 3.35) Kx,y) k dσ x) = O r k To verify 3.35), we write for all y. 2 x 2 x y ) = x y 2 + x 2 y 2. Therefore, it leads us to verify that x 2 y ) x y 6 C for some constant independent of y. By the homogeneity of the integr, we only need to consider, for any y, x 2 y ) x y 6 dσ x) C B It is easily seen that 3.37) holds if y δ or y δ, for any given δ>0. But for y δ,),+δ), we have the following calculus inequality for any N 3: 3.38) dσ x) x y N C y N 3. B We have thus established 3.37), hence 3.35) 3.32). To prove 3.33), we note that, since / x y 2 is the Green s function for the Laplacian on,wehave r dσ x) = 2, when y <r; 3.39) S 3 r Sr 3 x y 2 y 2, when y >r. It follows that we can write w r r) = Kx,y)Fy)dy, where r 6π 2 y 2, when y >r; 3.40) Kx,y) = ) 2 y 2 6π 2 r r 2, when y <r.

16 538 CHANG, QING, AND YANG Thus, ) w 2 ) w 2 dσ x) r r F y)dy Kx,y) Kx,y) ) 2 dσ x) F y)dy. We again compute y 2 x 2) x y 2 y 2) Kx,y) Kx,y) 6π 2 x x y 2 y 2, for y > x ; = y 2 x 2) x y 2 x 2) 6π 2 x x y 2 x 2, for y < x. Therefore, 3.4) Kx,y) Kx,y) ) 2 dσ x) = O x 2, as x y > x /2, 3.42) Kx,y) Kx,y) C x x y 2 x 2 = o x Combining 3.40), 3.4), 3.42), we obtain ) w 2 ) w 2 dσ x) r r ) = O y > x /2 x 2 F y)dy + This establishes 3.33). Now we are ready to estimate V r) V 2 r). Lemma 3.5. exists is positive. Then ), as x y x /2. y x /2 o ) ) x 2 F y)dy = o x 2. Suppose that e 2w dx 2 is a normal metric on suppose that lim +r w ) r r 3.43) V 2 r) = V 2 r) +o) ), as r,

17 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS ) V r) = V r) +o) ), as r. Proof. Let us denote by a = w/ r)r), b = e w r). Then V 2 r) = ) 4 r +a b 2, V 2 r) V 2 r) = ) 4 r +a e 2w b 2) + ) w e 4 r a 2w b 2). Applying Lemmas , we have ) V 2 r) V 2 r) = r +a b 2 o) 3.45) + ) w 2 ) /2 r a e 2w b 2) ) /2 2 = V 2 r)o)+ r b2 o), which implies 3.43) under the assumption that lim r +ra) > 0. Similarly, V r) = ) 2 4 r +a b, V r) V r) = ) 2 4 r +a e w b ) + ) w e 2r r a w b ) + ) w 2 ) a 2 e w. 4 r Applying Hölder inequality Lemmas , we again obtain V r) V r) = V r)o)+ b r 2 o), which again implies 3.44) under the assumption that lim r + ra) > 0. So the lemma is proved. We now discuss the metric that is an average of the metric e 2w dx 2 on over the spheres 0). For convenience, we would use the cylindrical coordinates again.

18 540 CHANG, QING, AND YANG Then wr) = we t ), vt) = wr)+t. As we have verified in Section 2, v satisfies the following: 3.46) with v 4v 2 = Qe 4v dσ x) = Ft), 0) 0) <t<, 3.47) Ft)dt = 2 S 3 Qe 4w dx = π 2 Qe 4w dx < 3.48) Ft) dt π 2 Q e 4w dx <. Lemma 3.6. Suppose that,e 2w dx 2 ) is a complete normal metric. Then its averaged metric,e 2 wr) dx 2 ) is also a complete metric. Proof. This is basically a consequence of Lemma 3.2. By the argument given in the proof of Lemma 3.5 we have 3.49) S 3 e wrσ) dσ = e wr) e o), S 3 where o) 0asr.So r r S 3 e wrσ) r 3.50) drdσ = S 3 r 0 r 0 S 3 e wrσ) dσ dr = e wr) e o) dr, S 3 r 0 which proves the lemma. Recall that in the statement of Theorem 2.7 we have assumed that the sign of the scalar curvature is positive, in order to establish the fact that the coefficient of the e 2t term vanishes, in the expression 2.23) of the conformal factor v. We have also pointed out in Remark 2.6 that to achieve the same purpose, it suffices to prove that v t) = O), ast tends to infinity. We will now establish this later fact for the average metric of a normal metric. Lemma 3.7. Suppose that,e 2w dx 2 ) is a normal metric. Then 3.5) Proof. 3.52) We compute wr) = S 3 wr) C r 2. S 3 wrσ)dσ = S 3 r S 3 r wx)dσ x).

19 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 54 Then, by the assumption that the metric is normal, we have wr) = { } ) 4π 2 Sr 3 dσ x) Qy)e 4wy) dy. x y 2 Thus, it follows from 3.40) 3.53) that wr) 3.54) 2π 2 r 2 Q e 4w dy. We have thus established the lemma. Theorem 3.8. Suppose that,e 2w dx 2 ) is a complete normal metric. Then Qe 4w dx 4π 2, 3.55) Furthermore, if 3.56) then 3.57) lim r for all l, m 4. 0) e3w dσ x) ) 4/3 B r 0) e4w dx S 3 r = 4π 2 Qe 4w dx 0. ) 4π 2 Qe 4w dx > 0, lim C l,mr) = r 4π 2 Qe 4w dx, Proof. We follow the outline indicated at the beginning of this section. We first look at the averaged metric e 2 wr) dx 2. Applying Lemmas Remark 2.6 in the last section, we have lim r w 3.58) r) = r r 2 S 3 Qe 4w dx. We then apply Corollary 3.3 get 3.59) lim r 0) e3w dσ x) ) 4/3 B r 0) e4w dx = lim r S 3 4/3 e 4 w r 4. Vr) Thus, it follows from 3.30) 3.3), with a proof similar to the proof of Theorem 2.7, that 3.55) holds. By the same reasoning, 3.57) follows from Lemma 3.5 under the condition of 3.56).

20 542 CHANG, QING, AND YANG Remark3.9. In the statements of Theorem 3.8, we have established 3.55) without the additional assumption that 3.56) holds. But we have established the further isoperimetric equalities 3.57) for higher order mixed volumes only under the additional assumption 3.56). In view of the fact that the classical isoperimetric inequalities 2.5) hold in general only for convex bodies for domains in R n, it remains an interesting problem to see if 3.57) holds even in the degenerate case when the constant in 3.56) is zero. 4. Proof of main theorems. In this section we show that a metric,e 2w dx 2 ) is a normal metric if its scalar curvature is nonnegative at infinity. It follows from the result in the last section that for a fairly large class of conformal metrics on we have the generalized Chern-Gauss-Bonnet formula 3.55) 3.57) in Theorem 3.8. Before we state prove our main result, we first remark that in general a complete conformal metric with integrable Paneitz curvature on may not be a normal metric. To see this, we can use the same example that was discussed at the beginning of Section 2. Recall that, in cylindrical coordinate, denote v = e 2t +t; then,e 4v dx 2 ) is a complete metric with Paneitz curvature Q = 0. It is also clearly not a normal metric. This justifies the additional assumption in the statement of Theorem 4.. Theorem 4.. Suppose the metric,e 2w dx 2 ) is a complete metric with the Paneitz curvature Q integrable, the scalar curvature is nonnegative at infinity. Then it is a normal metric. Proof. First denote vx) = 3 y 4.) 2π 2 log x y Qe4w dy, let h = w v. We will now show that the biharmonic function h on is a constant. Recall the transformation formula for the scalar curvature, 4.2) w + w 2 = R 6 e2w, where R is the scalar curvature for the metric,e 2w dx 2 ). Notice that h is a harmonic function. Thus, 4.3) hx 0 ) = x 0 ) = x 0 ) x 0 ) x 0 ) hdσ w 2 + R ) dσ 6 x 0 ) x 0 ) v dσ. The first term on the right of 4.3) is nonpositive when r is large enough by our assumption that R is nonnegative. We now observe that, by an argument similar to

21 CHERN-GAUSS-BONNET INTEGRAL FOR CONFORMAL METRICS 543 the proof of Lemma 3.7 in Section 3, we have v dσ = 3 { π 2 S 3 4.4) x 0 ) 3 π 2 r 2 Q e 4w dy. S 3 rσ +x 0 y 2 dσ Therefore, by taking r, we have, for each x 0, } Qy)e 4wy) dy. 4.5) hx 0 ) 0. Thus, h = C 0 for some nonpositive constant by Liouville theorem for harmonic functions. Thus, any partial derivative of h is harmonic; that is, 4.6) h xi = 0. Applying the mean value theorem again, we have 2 h xi x 0 ) 2 = h xi dσ 4.7) x 0 ) h 2 dσ. x 0 ) x 0 ) x 0 ) But 4.8) h 2 2 w 2 +2 v 2 = 2C 0 R 3 e2w +2 v 2 4.9) ) v 2 C x y 2 Q e4w dy Similarly, we conclude that, for each x 0, Q e 4w dy ) C x y 2 Q e4w dy. 4.0) h xi x 0 ) 2 2C 0, which implies that all partial derivatives of h are constants. Then h = C 0 = 0, which finally implies that all partial derivatives of h vanish by 4.0). Thus h is a constant. Main theorem. Suppose that,e 2w dx 2 ) is a complete metric with its Paneitz curvature absolutely integrable, its scalar curvature is nonnegative at infinity. Then conclusions in Theorem 3. hold. In particular, 4.) lim r 0) e3w dσ x) ) 4/3 42π 2 ) /3 B r 0) e4w dx = 4π 2 Qe 4w dx 0.

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