P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (119–146)

CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE

S-Y A C

Abstract This is the lecture notes for the author’s Emmy Noether lecture at 2018, ICM, Rio de Janeiro, Brazil. It is a great honor for the author to be invited to give the lecture.

Introduction

In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral conformal invariants on 4-manifolds and geometric applications. The development was heavily influenced by many earlier pioneer works; recent progress in conformal geometry has also been made in many different directions, here we will only present some slices of the development. The notes is organized as follows. In section 1, we briefly describe the prescribing Gaussian curvature problem on com- pact surfaces and the Yamabe problem on n-manifolds for n 3; in both cases some  second order PDE have played important roles. In section 2, we introduce the quadratic curvature polynomial 2 on compact closed 4- manifolds, which appears as part of the integrand of the Gauss–Bonnet–Chern formula. We discuss its algebraic structure, its connection to the 4-th order Paneitz operator P4 and its associated 4-th order Q curvature. We also discuss some variational approach to study the curvature and as a geometric application, results to characterize the diffeomor- 4 2 phism type of (S ; gc) and (CP ; gFS ) in terms of the size of the conformally invariant quantity: the integral of 2 over the manifold. In section 3, we extend our discussion to compact 4-manifolds with boundary and introduce a third order pseudo-differential operator P3 and 3-order curvature T on the boundary of the manifolds. In section 4, we shift our attention to the class of conformally compact Einstein (ab- breviated as CCE) four-manifolds. We survey some recent research on the problem of “filling in” a given 3-dimensional manifold as the conformal infinity of a CCE manifold. We relate the concept of ”renormalized” volume in this setting again to the integral of 2. ICM Emmy Noether Lecture ICM 2018, Rio de Janeiro, Brazil. Research of Chang is supported in part by NSF Grant DMS-1509505.

119 120 SUN-YUNG ALICE CHANG

In section 5, we discuss some partial results on a compactness problem on CCE manifolds. We believe the compactness results are the key steps toward an existence theory for CCE manifolds. The author is fortunate to have many long-term close collaborators, who have greatly contributed to the development of the research described in this article – some more than the author. Among them Matthew Gursky, Jie Qing, Paul Yang and more recently Yuxin Ge. She would like to take the chance to express her deep gratitude toward them, for the fruitful collaborations and for the friendships.

1 Prescribing Gaussian curvature on compact surfaces and the Yamabe problem

In this section we will describe some second order elliptic equations which have played important roles in conformal geometry. On a compact surface (M; g) with a Riemannian metric g, a natural curvature invari- ant associated with the Laplace operator ∆ = ∆g is the Gaussian curvature K = Kg . 2w Under the conformal change of metric gw = e g, we have

2w (1.1) ∆w + K = Kg e on M: w The classical uniformization theorem to classify compact closed surfaces can be viewed as finding solution of equation (1.1) with Kg 1, 0, or 1 according to the sign of w Á R Kg dvg . Recall that the Gauss–Bonnet theorem states

Z (1.2) 2 (M ) = Kgw dvgw ; M where (M ) is the Euler characteristic of M , a topological invariant. The variational functional with (1.1) as Euler equation for Kgw = constant is thus given by Moser’s functional (Moser [1970/71, 1973])

Z 2 Z ÂZ Ã RM dvgw (1.3) Jg [w] = w dvg + 2 Kwdvg Kdvg log : M jr j M M RM dvg There is another geometric meaning of the functional J which influences the later development of the field, that is the formula of Polyakov [1981]

det ( ∆) ! (1.4) J [w] = 12 log g g det ( ∆) gw for metrics gw with the same volume as g; where the determinant of the Laplacian det ∆g is defined by Ray–Singer via the “regularized” zeta function. In Onofri [1982], (see also Hong [1986]), Onofri established the sharp inequality that on the 2-sphere J [w] 0 and J [w] = 0 precisely for conformal factors w of the form 2w  e g0 = T g0 where T is a Moebius transformation of the 2-sphere. Later Osgood, Phillips, and Sarnak [1988a,b] arrived at the same sharp inequality in their study of CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 121 heights of the Laplacian. This inequality also plays an important role in their proof of the C 1 compactness of isospectral metrics on compact surfaces. n On manifolds (M ; g) for n greater than two, the conformal Laplacian Lg is defined n 2 as Lg = ∆g + cnRg where cn = , and Rg denotes the scalar curvature of the 4(n 1) metric g. An analogue of equation (1.1) is the equation, commonly referred to as the Yamabe equation (1.5), which relates the scalar curvature under conformal change of metric to the background metric. In this case, it is convenient to denote the conformal 4 ˆ n 2 metric as g = u g for some positive function u, then the equation becomes

n+2 ˆ n 2 (1.5) Lg u = cn R u :

The famous Yamabe problem to solve (1.5) with Rˆ a constant has been settled by Yam- abe [1960], Trudinger [1968], Aubin [1976] and Schoen [1984]. The corresponding problem to prescribe scalar curvature has been intensively studied in the past decades by different groups of mathematicians, we will not be able to survey all the results here. We will only point out that in this case the study of Rˆ = c, where c is a constant, over class of metrics gˆ in the conformal class [g] with the same volume as g, is a variational problem with respect to the functional Fg [u] = R n Rgˆdvgˆ for any n 3. Again the M  sign of the constant c agrees with the sign of the Yamabe invariant

RM Rgˆdvgˆ (1.6) Y (M; g) := inf n 2 : gˆ [g] n 2 (vol gˆ)

2 2 curvature on 4-manifold

2.1 Definition and structure of 2. We now introduce an integral conformal invari- ant which plays a crucial role in this paper, namely the integral of 2 curvature on four- manifolds. To do so, we first recall the Gauss–Bonnet–Chern formula on closed compact mani- fold (M; g) of dimension four:

2 Z 1 2 Z 1 2 2 (2.7) 8 (M ) = Wg dvg + (Rg 3 Ricg )dvg ; M 4j j M 6 j j where (M ) denotes the Euler characteristic of M , Wg denotes the Weyl curvature, Rg the scalar curvature and Ricg the Ricci curvature of the metric g. In general, the Weyl curvature measures the obstruction to being conformally flat. More precisely, for a manifold of dimension greater or equals to four, Wg vanishes in a neighborhood of a point if an only if the metric is locally conformal to a Euclidean metric; i.e., there are local coordinates such that g = e2w dx 2 for some function w. j jn Thus for example, the standard round metric gc on the sphere S has Wg 0. c Á In terms of conformal geometry, what is relevant to us is that Weyl curvature is a pointwise conformal invariant, in the sense that under conformal change of metric gw = 2w 2w 2 2 e g, Wg = e Wg , thus on 4-manifold Wg dvg = Wg dvg ; this implies j w j j j j w j w j j 122 SUN-YUNG ALICE CHANG in particular that the first term in the Gauss–Bonnet–Chern formula above

Z 2 g W g dvg ! M j j is conformally invariant. For reason which will be justified later below, we denote

1 2 2 (2.8) 2(g) = (R 3 Ricg ) 6 g j j and draw the conclusion from the above discussion of the Gauss–Bonnet–Chern formula that Z g 2(g)dvg ! M is also an integral conformal invariant. This is the fundamental conformal invariant which will be studied in this lecture notes. We begin by justifying the name of “2” curvature. On manifolds of dimensions greater than two, the Riemannian curvature tensor Rm can be decomposed into the different components. From the perspective of conformal geometry, a natural basis is the Weyl tensor W , and the Schouten tensor, defined by

R Ag = Ricg g: 2(n 1) The curvature tensor can be decomposed as

1 Rmg = Wg Ag g: ˚ n 2 ^ 2w Under conformal change of metrics gw = e g, since the Weyl tensor W transforms by scaling, only the Schouten tensor depends on the derivatives of the conformal factor. It is thus natural to consider k(Ag ), the k-th symmetric function of the eigenvalues of the Schouten tensor Ag , as curvature invariants of the conformal metrics. n 2 When k = 1, 1(Ag ) = T rg Ag = 2(n 1) Rg , so the 1-curvature is a dimensional multiple of the scalar curvature. 1 2 2 When k = 2, 2(Ag ) = P i j = ( T rg Ag Ag ), where the s are the i

1 2 2 2(g) = 2(Ag ) = (R 3 Ricg ): 6 g j j

When k = n, n(Ag ) = determinant of Ag , an equation of Monge–Ampère type. In view of the Yamabe problem, it is natural to ask the question under what condition can one find a metric gw in the conformal class of g, which solves the equation

(2.9) 2(Agw ) = constant: CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 123

To do so, we first observe that as a differential invariant of the conformal factor w,

k(Agw ) is a fully nonlinear expression involving the Hessian and the gradient of the conformal factor w. We have

2 2 w Ag = (n 2) w + dw dw jr j + Ag : w fr ˝ 2 g To illustrate that (2.9) is a fully non-linear equation, we have when n = 4,

4w 2 2 2 2(Ag )e = 2(Ag ) + 2((∆w) w w jr j (2.10) + ∆w w 2 + ( w; w 2)) jr j r rjr j +lower order terms: where all derivative are taken with respect to the g metric. For a symmetric n n matrix M , we say M Γ+ in the sense of Gȧrding [1959] if  2 k k(M ) > 0 and M may be joined to the identity matrix by a path consisting entirely of matrices Mt such that k(Mt ) > 0. There is a rich literature concerning the equation

2 (2.11) k( u) = f; r for a positive function f , which is beyond the scope of this article to cover. Here we will only note that when k = 2,

2 2 2 2 (2.12) 2( u) = (∆u) u = f; r jr j for a positive function f . We remark the leading term of equations (2.12) and (2.10) agree. We now discuss a variational approach to study the equation (2.9) for 2 curvature .

Recall in section 1 we have mentioned that the functional F (g) := RM n Rg dvg is variational in the sense that when n 3 and when one varies g in the same conformal  class of metrics with fixed volume, the critical metric when attained satisfies Rgˆ Á constant; while when n = 2, this is no longer true with Rg replaced by Kg and one needs to replace the functional F (g) by the Moser’s functional Jg . Parallel phenomenon happens when one studies the 2 curvature. It turns out that when n > 2 and n 4, the functional F2(g) := R n 2(g)dvg is variational when ¤ M one varies g in the same conformal class of metrics with fixed volume, while this is no longer true when n = 4. In section 2a below, we will describe a variational approach to study equation (2.9) in dimension 4 and the corresponding Moser’s functional. Before we do so, we would like to end the discussion of this section by quoting a result of Gursky and Viaclovsky [2001]. In dimension 3, one can capture all metrics with constant sectional curvature (i.e. space forms) through the study of 2. Theorem 2.1. (Gursky and Viaclovsky [ibid.]) On a compact 3-manifold, for any Rie- mannian metric g, denote F2(g) = R 2(Ag )dvg . Then a metric g with F2(g) 0 is M  critical for the functional F2 restricted to class of metrics with volume one if and only if g has constant sectional curvature. 124 SUN-YUNG ALICE CHANG

2.2 4-th order Paneitz operator, Q-curvature. We now describe the rather surpris- ing link between a 4-th order linear operator P4, its associated curvature invariant Q4, and the 2-curvature. We first recall on (M n; g), n 3, the second order conformal Laplacian operator n 2  4 = ∆ + ˆ = n 2 L 4(n 1) R transforms under the conformal change of metric g u g, as n+2 4 (2.13) Lgˆ(') = u n 2 Lg (u ' ) for all ' C 1(M ): 2 There are many operators besides the Laplacian ∆ on compact surfaces and the con- formal Laplacian L on general compact manifold of dimension greater than two which have the conformal covariance property. One class of such operators of order 4 was studied by Paneitz [2008] (see also Eastwood and Singer [1985]) defined on (M n; g) when n > 2; which we call the conformal Paneitz operator: n 4 (2.14) P n = ∆2 + ı (a Rg + b Ric) d + Qn 4 n n 2 4 and

n 2 2 1 (2.15) Q = cn Ric + dnR ∆R; 4 j j 2(n 1) where an, bn, cn and dn are some dimensional constants, ı denotes the divergence, d the deRham differential. The conformal Paneitz operator is conformally covariant. In this case, we write the 4 conformal metric as gˆ = u n 4 g for some positive function u, then for n 4, ¤ n n+4 n 4 (2.16) (P ) (') = u n 4 (P ) (u ' ) for all ' C 1(M ): 4 gˆ 4 g 2 n n Properties of (P4 ;Q4 ) have been intensively studied in recent years, with many sur- prisingly strong results. We refer the readers to the recent articles (Case and P. Yang [2013], Gursky and Malchiodi [2015], Gursky, Hang, and Lin [2016], Hang and P. C. Yang [2015], Hang and P. C. Yang [2016] and beyond).

n n 4 n Notice that when n is not equal to 4, we have P4 (1) = 2 Q4 , while when n = 4, 4 4 one does not read Q4 from P4 ; it was pointed out by T. Branson that nevertheless both 4 4 P := P4 and Q := Q4 are well defined (which we named as Branson’s Q-curvature): 2 P' := ∆2' + ı  Rg 2Ricà d': 3 The Paneitz operator P is conformally covariant of bidegree (0; 4) on 4-manifolds, i.e.

4! 4 Pg (') = e Pg (') for all ' C 1(M ) : w 2 The Q curvature associated with P is defined as

1 1 2 2 (2.17) 2Qg = ∆Rg + (R 3 Ricg )): 6 6 g j j CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 125

Thus the relation between Q curvature and 2 curvature is 1 (2.18) 2Qg = ∆Rg + 2(Ag ): 6 The relation between P and Q curvature on manifolds of dimension four is like that of the Laplace operator ∆ and the Gaussian curvature K on compact surfaces. 4w (2.19) Pg w + 2Qg = 2Qgw e :

Due to the pointwise relationship between Q and 2 in (2.18), notice on compact 1 closed 4-manifolds M , RM ∆Rg dvg = 0, thus RM Qg dvg = 2 R 2(Ag )dvg is also an integral conformal invariant. But Q curvature has the advantage of being prescribed by the linear operator P in the equation (2.19), which is easier to study. We remark the situation is different on compact 4-manifolds with boundary which we will discuss in later part of this article

Following Moser, the functional to study constant Qg metric with gw [g] is given w 2 by

R e4w dv II [w] = P w; w + 4 Z Qwdv ÂZ Qdvà log : h i R dv

In view of the relation between Q and 2 curvatures, if one consider the variational functional III whose Euler equation is ∆R = constant ,

1 ÂZ 2 Z 2 Ã III [w] = R dvg R dv ; 3 gw w and define 1 F [w] = II [w] III [w]; 12 we draw the conclusion: Proposition 2.2. (Chang and P. C. Yang [2003], Chang, Gursky, and P. C. Yang [2002], see also Brendle and Viaclovsky [2004] for an alternative approach.) F is the Lagrangian functional for 2 curvature. We remark that the search for the functional F above was originally motivated by the study of some other variational formulas, e.g. the variation of quotients of log de- terminant of conformal Laplacian operators under conformal change of metrics on 4- manifolds, analogues to that of the Polyakov formula (1.4) on compact surfaces . We refer the readers to articles (Branson and Ørsted [1991], Branson, Chang, and P. C. Yang [1992], Chang, Gursky, and P. C. Yang [2002], Okikiolu [2001]) on this topic. We also remark that on compact manifold of dimension n, there is a general class of conformal covariant operators P k of order 2k, for all integers k with 2k n. This is 2 Ä the well-known class of GJMS operators Graham, Jenne, Mason, and Sparling [1992], where P2 coincides with the conformal Laplace operator L and where P4 coincides with the 4-th order Paneitz operator. GJMS operators have played important roles in many recent developments in conformal geometry. 126 SUN-YUNG ALICE CHANG

2.3 Some properties of the conformal invariant R 2. We now concentrate on the class of compact, closed four manifolds which allow a Riemannian metric g in the class A, where

 Z  A := g ;Y (M; g) > 0; 2(Ag )dvg > 0 : j M Notice that for closed four manifolds, it follows from equation (2.18) that

Z Z 2 Qg dvg = 2(Ag )dvg ; M M so in the definition of A, we could also use Q instead of 2. We recall some important properties of metrics in A. Theorem 2.3. (Chang and P. C. Yang [1995], Gursky [1999], Chang, Gursky, and P. C. Yang [2002])

2 1. If Y (M; g) > 0, then RM 2(g)dvg 16 , equality holds if and only if (M; g) 4 Ä is conformally equivalent to (S ; gc). 2. g A, then P 0 with kernel (P ) consists of constants; it follows there exists 2  some gw [g] with Qg = constant and Rg > 0. 2 w w 3. g A, then there exists some gw [g] with 2(Agw ) > 0 and Rgw > 0; i. e. 2 + 2 gw exists in the positive two cone Γ2 of Ag in the sense of Gȧrding [1959]. We remark + g Γ implies Ricg > 0, as a consequence the first Betti number b1 of M is zero. 2 2 4. g A, then there exists some gw [g] with 2(Agw ) = 1 and Rgw > 0: 2 2 4 5. When (M; g) is not conformally equivalent to (S ; gc) and g A, then for any 2 positive smooth function f defined on M , there exists some gw [g] with 2(Ag ) = 2 w f and Rgw > 0:

We remark that techniques for solving the 2 curvature equation can be modified to 2 solve the equation 2 = 1 + c W for some constant c, which is the equation we will j j use later in the proof of the theorems in section 2d.

As a consequence of above theorem we have

4 Corollary 2.1. On (M ; g), g A if and only if there exists some gw [g] with + 2 2 gw Γ . 2 2 A significant result in recent years is the following “uniqueness” result.

Theorem 2.4. (Gursky and Streets [2018]) 4 4 + + Suppose (M ; g) is not conformal to (S ; gc) and g Γ , then gw [g] with gw Γ 2 2 2 2 2 and with 2(Agw ) = 1 is unique. + The result was established by constructing some norm for metrics in Γ2 , with respect to which the functional F is convex. The result is surprising in contrast with the famous 1 n example of Schoen [1989] where he showed that on (S S ; gprod ), where n 2,   the class of constant scalar curvature metrics (with the same volume) is not unique. CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 127

2.4 Diffeomorphism type. In terms of geometric application, this circle of ideas may be applied to characterize the diffeomorphism type of manifolds in terms of the the rel- ative size of the conformal invariant R 2(Ag )dVg compared with the Euler number of the underlying manifold, or equivalently the relative size of the two integral conformal 2 invariants R 2(Ag )dvg and R W dvg . jj jjg Note: In the following, we will view the Weyl tensor as an endomorphism of the space of two-forms: W :Ω2(M ) Ω2(M ). It will therefore be natural to use the ! norm associated to this interpretation, which we denote by using . In particular, k
k W 2 = 4 W 2: j j k k Theorem 2.5. (Chang, Gursky, and P. C. Yang [2003]) Suppose (M; g) is a closed 4-manifold with g A. 2 2 4 4 (a). If RM W g dvg < RM 2(Ag )dvg then M is diffeomorphic to either S or RP . jj jj 4 4 2 (b). If M is not diffeomorphic to S or RP and RM W g dvg = RM 2(Ag )dvg , 2 jj jj then (M; g) is conformally equivalent to (CP ; gFS ). Remark 1: The theorem above is an L2 version of an earlier result of Margerin [1998]. The first part of the theorem should also be compared to an result of Hamilton [1986]; where he pioneered the method of Ricci flow and established the diffeomorphism of M 4 to the 4-sphere under the assumption when the curvature operator is positive. Remark 2: The assumption g A excludes out the case when (M; g) = (S 3 1 2  S ; gprod ), where W g = 2(Ag ) 0. jj jj Á Sketch proof of Theorem 2.5. Proof. For part (a) of the theorem, we apply the existence argument to find a conformal metric gw which satisfies the pointwise inequality (2.20) 2 2 Wg < 2(Ag ) or 2(Ag ) = Wg + c for some constant c > 0: jj w jj w w jj w jj The diffeomorphism assertion follows from Margerin [1998] precise convergence result for the Ricci flow: such a metric will evolve under the Ricci flow to one with constant curvature. Therefore such a manifold is diffeomorphic to a quotient of the standard 4- sphere.

For part (b) of the theorem, we argue that if such a manifold is not diffeomorphic to 2 the 4-sphere, then the conformal structure realizes the minimum of the quantity R Wg dvg , j j and hence its Bach tensor vanishes; i.e.

k l 1 kl Bg = Wikjl + R Wikjl = 0: r r 2 As we assume g A, we can solve the equation 2 2 (2.21) 2(Ag ) = (1 ) Wg + c; w jj w jj where c is a constant which tends to zero as  tends to zero. We then let  tends to zero. We obtain in the limit a C 1;1 metric which satisfies the equation on the open set 128 SUN-YUNG ALICE CHANG

Ω = x W (x) 0 : f j ¤ g 2 (2.22) 2(Ag ) = Wg : w jj w jj

We then decompose the W eyl curvature of gw into its self dual and anti-self dual part as in the Singer–Thorpe decomposition of the full curvature tensor, apply the Bach equa- tion to estimate the operator norm of each of these parts as endormorphism on curvature tensors, and reduce the problem to some rather sophisticated Lagrange multiplier prob- lem. We draw the conclusion that the curvature tensor of gw agrees with that of the Fubini–Study metric on the open set Ω. Therefore Wg is a constant on Ω, thus W j w j cannot vanish at all. From this, we conclude that gw is Einstein (and under some pos- itive orientation assumption) with Wgw = 0. It follows from a result of Hitchin (see Besse [1987], chapter 13) that the limit metric gw agrees with the Fubini–Study metric of CP 2

We now discuss some of the recent joint work of M. Gursky, Siyi Zhang and myself Chang, Gursky, and Zhang [2018] extending the theorem 2.5 above to a perturbation theorem on CP 2.

For this purpose, for a metric g A, we define the conformal invariant constant 2 ˇ = ˇ([g]) defined as

Z 2 Z W g dvg = ˇ 2(Ag )dvg : jj jj M

Lemma 2.1. Given g A, if 1 < ˇ < 2, then M 4 is either homeomorphic to either 4 4 + 2 4 2 + S or RP (hence b2 = b2 = 0) or M is homeomorphic to CP (hence b2 = 1).

We remark that ˇ = 2 for the product metric on S 2 S 2.  An additional ingredient to establish the lemma above is the Signature formula :

2 Z + 2 2 12  = ( W W )dv; M jj jj jj jj

+ + where  = b2 b2, W is the self dual part of the Weyl curvature and W jj+ jj jj jj the anti-self-dual part, b2 , b2 the positive and negative part of the intersection form; together with an earlier result of Gursky [1998].

In view of the statement of Theorem 2.5 above, it is tempting to ask if one can change the “homeomorphism type” to “diffeomorphism type” in the statement of the Lemma. So far we have not been able to do so, but we have a perturbation result. Theorem 2.6. (Chang, Gursky, and Zhang [2018]) There exists some  > 0 such that + if (M; g) is a four manifold with b2 > 0 and with a metric of positive Yamabe type satisfying with 1 < ˇ([g]) < 1 + , then (M; g) is diffeomorphic to standard CP 2. CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 129

Sketch proof of Theorem 2.6. + Proof. A key ingredient is to apply the condition b2 > 0 to choose a good representa- tive metric gGL [g], which is constructed in the earlier work of Gursky [2000] and 2 used in Gursky and LeBrun [1998] and Gursky and Lebrun [1999]). To do so, they considered a generalized Yamabe curvature

+ R˜g = Rg 2p6 W g ; jj jj + and noticed that on manifold of dimension 4, due to the conformal invariance of W g , jj jj the corresponding Yamabe type functional ˜ RM Rgw dvgw g g := inf 1 ! gw [g] 2 2 (volgw ) still attains its infimum; which we denote by gGL. The key observation in Gursky [2000] + ˜ is that b2 > 0 implies g < 0 (thus RGL < 0). To see this, we recall the Bochner formula satisfied by the non-trivial self dual harmonic 2-form  at the extreme metric: 1 1 ∆(  2) =  2 2W + < ;  > + R  2; 2 j j jr j 3 j j which together with the algebraic inequality that 1 1 2W + < ;  > + R  2 R˜  2; 3 j j  3 j j forces the sign of R˜GL when  is non-trivial. To continue the proof of the theorem, we notice that for a given metric g satisfying the conformal pinching condition 1 < ˇ([g]) < 1+ on its curvature, the corresponding gGL would satisfy G2(gGL) C (), where for k = 2; 3; 4, Ä Z ¯ k 0 k k ˜ k Gk(g) := (R R) + Ric + W + R Á dvg ; M j j jj jj j j where R¯ denotes the average of the scalar curvature R over the manifold, Ric0 the denote the traceless part of the Ricci curvature and R˜ the negative part of R˜, and where C () is a constant which tends to zero as  tends to zero. We now finish the proof of the theorem by a contradiction argument and by applying the Ricci flow method of Hamilton @ g(t) = 2Ricg(t) @t to regularize the metric gGL. Suppose the statement of the theorem is not true, let gi be a sequence of metrics f g satisfying 1 < ˇ([gi ]) < 1 + i with i tends to zero as i tends to infinity. Choose gi (0) = (gi )GL to start the Ricci flow, we can derive the inequality @ G2(g(t)) aG2(gt ) bG4(g(t)) @t Ä 130 SUN-YUNG ALICE CHANG

for some positive constants a and b, also at some fixed time t0 independent of  and i for each gi . We then apply the regularity theory of parabolic PDE to derive that some sub- 2 sequences of gi (t0) converges to the gFS metric of CP , which in turn implies the f g original subsequence (M; gi (0) ) hence a subsequence of (M; gi ) is diffeomorphic 2 f g f g to (CP ; gFS ). We thus reach a contradiction to our assumption. The reader is referred to the preprint Chang, Gursky, and Zhang [2018] for details of the proof. We end this section by pointing out there is a large class of manifolds with metrics in the class A. By the work of Donaldson–Freedman (see Donaldson [1983], Freedman [1982]) and Lichnerowicz vanishing theorem, the homeomorphism type of the class of simply-connected 4-manifolds which allow a metric with positive scalar curvature consists of S 4 together with kCP 2#lCP 2 and k(S 2 S 2). Apply some basic algebraic  manipulations with the Gauss–Bonnet–Chern formula and the Signature formula, we can show that a manifold which admits g A satisfies 4 + 5l > k. The round metric 2 on S 4, the Fubini–Study metric on CP 2, and the product metric on S 2 S 2 are clearly  in the class A. When l = 0, which implies k < 4, the class A also includes the metrics constructed by LeBrun, Nayatani, and Nitta [1997] on kCP 2 for k 2. When l = 1, Ä which implies k < 9, the class A also includes the (positive) Einstein metric constructed by Page [1978] on CP 2#CP 2, the (positive) Einstein metric by Chen, Lebrun, and Weber [2008] on CP 2#2CP 2, and the Kähler Einstein metrics on CP 2#lCP 2 for 3 Ä l 8 as in the work of Tian [2000]. It would be an ambitious program to locate the Ä entire class of 4-manifolds with metric in A, and to classify their diffeomorphism types by the (relative) size of the integral conformal invariants discussed in this lecture.

3 Compact 4-manifold with boundary, (Q;T ) curvatures

To further develop the analysis of the Q-curvature equation, it is helpful to consider the associated boundary value problems. In the case of compact surface with boundary (X 2;M 1; g), where the metric g is defined on X 2 M 1; the Gauss–Bonnet formula [ becomes

(3.23) 2 (X) = Z K dv + I k d; X M where k is the geodesic curvature on M . Under conformal change of metric gw on X, the geodesic curvature changes according to the equation @ (3.24) w + k = k ew on M: @n gw One can generalize above results to compact four manifold with boundary 4 3 @ (X ;M ; g); with the role played by ( ∆; ) replaced by (P4;P3) and with (K; k) @n curvature replaced by (Q;T ) curvatures; where P4 is the Paneitz operator and Q the curvature discussed in section 2; and where P3 is some 3rd order boundary operator con- structed in Chang and Qing [1997a,b]. The key property of P3 is that it is conformally covariant of bidegree (0; 3), i.e.

3w (P3)gw = e (P3)g CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 131 when operating on functions defined on the boundary of compact 4-manifolds; and un- 2w 4 3 der conformal change of metric gw = e g on X we have at the boundary M

3w (3.25) P3w + T = Tgw e :

The precise formula of P3 is rather complicated (see Chang and Qing [1997a]). Here we will only mention that on (B4;S 3; dx 2), where B4 is the unit ball in R4, we have j j

2 Â1 @ @ à (3.26) P4 = ( ∆) ;P3 = ∆ + ∆˜ + ∆˜ and T = 2; 2 @n @n where ∆˜ the intrinsic boundary Laplacian on S 3. In general the formula for T curvature is also lengthy,

1 @ 1 1 3 1 3 1 ˜ T = R + RH R˛nˇnL˛ˇ + H TrL ∆H; 12 @n 6 9 3 3 where L is the second fundamental form of M in (X; g), and H the mean curvature, and n its the outside normal. In terms of these curvatures, the Gauss–Bonnet–Chern formula can be expressed as:

(3.27) 82 (X) = Z ( W 2 + 2Q) dv + I (L + 2T ) d: X jj jj M where L is a third order boundary curvature invariant that transforms by scaling under conformal change of metric, i.e. Ld is a pointwise conformal invariant. The property which is relevant to us is that

Z Qdv + I T d X M is an integral conformal invariant. It turns out for the cases which are of interest to us later in this paper, (X; g) is with totally geodesic boundary, that is, its second fundamental form vanishes. In this special case we have

1 @ (3.28) T = R: 12 @n

Thus in view of the definitions (2.13) and (3.28) of Q and T , in this case we have

ÂZ I Ã Z 2 Qdv + T d = 2 dv; X M X which is the key property we will apply later to study the renormalized volume and the compactness problem of conformal compact Einstein manifolds in sections 4 and 5 below. 132 SUN-YUNG ALICE CHANG

4 Conformally compact Einstein manifolds

4.1 Definition and basics, some short survey. Given a manifold (M n; [h]), when is it the boundary of a conformally compact Einstein manifold (X n+1; g+) with 2 + r g M = h? This problem of finding “conformal filling in” is motivated by problems j in the AdS/CFT correspondence in quantum gravity (proposed by Maldacena [1998]) and from the geometric considerations to study the structure of non-compact asymptot- ically hyperbolic Einstein manifolds. Here we will only briefly outline some of the progress made in this problem pertain- ing to the conformal invariants we are studying. Suppose that X n+1 is a smooth manifold of dimension n + 1 with smooth boundary @X = M n. A defining function for the boundary M n in X n+1 is a smooth function r on X¯n+1 such that 8 r > 0 in X; <ˆ r = 0 on M ; ˆdr 0 on M: : ¤ A Riemannian metric g+ on X n+1 is conformally compact if (X¯n+1; r2g+) is a com- pact Riemannian manifold with boundary M n for some defining function r. We denote 2 + h := r g M . j Conformally compact manifold (X n+1; g+) carries a well-defined conformal struc- ture on the boundary (M n; [h]) by choices of different defining functions r. We shall call (M n; [h]) the conformal infinity of the conformally compact manifold (X n+1; g+). If (X n+1; g+) is a conformally compact manifold and Ric[g+] = n g+, then we call (X n+1; g+) a conformally compact (Poincare) Einstein (abbreviated as CCE) man- ifold. We remark that on a CCE manifold X, for any given smooth metric h in the conformal infinity M , there exists a special defining function r (called the geodesic 2 + 2 defining function) so that r g M = h; and dr 2 + = 1 in a neighborhood of the j j jr g boundary [0; ) M , also the metric r2g+ has totally geodesic boundary. 

n+1 n Some basic examples. Example 1: On (B ;S ; gH)

2 2 ! Bn+1; Â Ã dy 2 : 1 y 2 j j j j n n+1 We can then view (S ; [gc]) as the compactification of B using the defining function

1 y r = 2 j j 1 + y j j 2 2 + 2 2  r à ! gH = g = r dr + 1 gc : 4 Example 2: AdS-Schwarzchild space

On (R2 S 2; g+);  m CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 133 where + 2 1 2 2 gm = Vdt + V dr + r gc;

2m V = 1 + r2 ; r

1 2 m is any positive number, r [rh; + ), t S () and gc the surface measure on S 2 21 2m2 and rh is the positive root for 1 + r r = 0. We remark, it turns out that in this + case, there are two different values of m so that both gm are conformal compact Einstein filling for the same boundary metric S 1() S 2. This is the famous non-unique “filling  in” example of Hawking and Page [1982/83].

Existence and non-existence results. The most important existence result is the “Am- bient Metric” construction by Fefferman and Graham [1985, 2012]. As a consequence of their construction, for any given compact manifold (M n; h) with an analytic metric h, some CCE metric exists on some tubular neighborhood M n (0; ) of M . This  later result was recently extended to manifolds M with smooth metrics by Gursky and Székelyhidi [2017]. A perturbation result of Graham and Lee [1991] asserts that in a smooth neighbor- n hood of the standard surface measure gc on S , there exist a conformal compact Einstein metric on Bn+1 with any given conformal infinity h. There is some recent important articles by Gursky–Han and Gursky–Han–Stolz (Gursky and Han [2017], Gursky, Han, and Stolz [2018]), where they showed that when X is spin and of dimension 4k 8, and the Yamabe invariant Y (M; [h]) > 0, then there  are topological obstructions to the existence of a Poincaré–Einstein g+ defined in the interior of X with conformal infinity given by [h]. One application of their work is that on the round sphere S 4k 1 with k 2, there are infinitely many conformal classes that  have no Poincaré–Einstein filling in in the ball of dimension 4k. The result of Gursky–Han and Gursky–Han–Stolz was based on a key fact pointed out Qing [2003], which relies on some earlier work of Lee [1995].

Lemma 4.1. On a CCE manifold (X n+1;M n; g+), assuming Y (M; [h]) > 0, there exists a compactification of g+ with positive scalar curvature; hence Y (X; [r2g+]) > 0.

Uniqueness and non-uniqueness results. Under the assumption of positive mass n+1 theorem, Qing [2003] has established (B ; gH ) as the unique CCE manifold with n (S ; [gc]) as its conformal infinity. The proof of this result was later refined and estab- lished without using positive mass theorem by Li, Qing, and Shi [2017] (see also Dutta and Dutta and Javaheri [2010]). Later in section 5 of this lecture notes, we will also prove the uniqueness of the CCE extension of the metrics constructed by Graham and Lee [1991] for the special dimension n = 3. As we have mentioned in the example 2 above, when the conformal infinity is S 1()  S 2 with product metric, Hawking and Page [1982/83] have constructed non-unique CCE fill-ins. 134 SUN-YUNG ALICE CHANG

4.2 Renormalized volume. We will now discuss the concept of “renormalized vol- ume” in the CCE setting, introduced by Maldacena [1998] (see also the works of Wit- ten [1998], Henningson and Skenderis [1998] and Graham [2000]). On CCE manifolds (X n+1;M n; g+) with geodesic defining function r, For n even, n n+2 Volg+ ( r >  ) = c0 + c2 + f g


2 1 + cn 2 + L log + V + o(1):  For n odd, n n+2 Volg+ ( r >  ) = c0 + c2 + f g




1 + cn 1 + V + o(1): We call the zero order term V the renormalized volume. It turns out for n even, L is 2 + independent of h [h] where h = r g M , and for n odd, V is independent of g [g], 2 j 2 and hence are conformal invariants. We recall

Theorem 4.1. (Graham and Zworski [2003], Fefferman and Graham [2002]) When n is even, I L = cn Qhdvh; M where cn is some dimensional constant.

Theorem 4.2. (Anderson [2001], P. Yang, King, and Chang [2006a], P. Yang, King, and Chang [2006b]) On conformal compact Einstein manifold (X 4;M 3; g+), we have

1 Z V = 2(Ag )dvg 6 X 4 for any compactified metric g with totally geodesic boundary. Thus

2 4 3 Z 2 8 (X ;M ) = W dvg + 6V: jj jjg

Remark: There is a generalization of Theorem 4.2 above for any n odd, with X 4 replaced n by X and with RX 4 2 replaced by some other suitable integral conformal invariants n+1 n+1 n + RXn+1 v on any CCE manifold (X ;M ; g ); see (Chang, Fang, and Graham [2014], also P. Yang, King, and Chang [2006a]).

Sketch proof of Theorem 4.2 for n = 3.

Lemma 4.2. (Fefferman and Graham [2002]) Suppose (X 4;M 3; g+) is conformally compact Einstein with conformal infinity (M 3; [h]), fix h [h] and r its corresponding geodesic defining function. Consider 2 4 (4.29) ∆g+ w = 3 on X ; CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 135 then w has the asymptotic behavior

w = log r + A + Br 3 near M , where A; B are functions even in r, A M = 0, and j Z V = B M : M j

Lemma 4.3. With the same notation as in Lemma 4.2, consider the metric g = gw = 2w + 1 @ e g , then g is totally geodesic on boundary with (1) Qg 0; (2) B M = Rg =   Á j 36 @n  1 T : 3 g

Proof of Lemma 4.3. :

+ + Proof. Recall we have g is Einstein with Ricg+ = 3g , thus

Pg+ = ( ∆g+ ) ( ∆g+ 2) ı and 2Qg+ = 6: Therefore

2w P + w + 2Q + = 0 = 2e Q : g g g Assertion (2) follows from a straight forward computation using the scalar curvature equation and the asymptotic behavior of w. Applying Lemmas 4.2 and 4.3, we get

1 @ 6V = 6 I B d = I R d M h g h M 3 j 6 M 3 @n = 2(Z Q + I T ) = Z  (A )dv : g g 2 g g X M X 4

For any other compactified metric g with totally geodesic boundary, RX 4 2(g)dvg is a conformal invariant, and V is a conformal invariant, thus the result holds once for g, holds for any such g in the same conformal class, which establishes Theorem 4.2.

5 Compactness of conformally compact Einstein manifolds on dimension 4

In this section, we will report on some joint works of Yuxin Ge and myself Chang and Ge [2018] and also Yuxin Ge, Jie Qing and myself Chang, Ge, and Qing [2018]. The project we work on is to address the problem of given a sequence of CCE man- 4 3 + 2 + ifolds (X ;M ; (gi ) ) with M = @X and gi = r (gi ) a sequence of com- f g f g f i g pactified metrics, denote hi = gi M , assume hi forms a compact family of metrics j f g in M , is it true that some representatives g¯i [gi ] with g¯i M = hi also forms a 2 f j g f g compact family of metrics in X? Let me mention the eventual goal of the study of the compactness problem is to show existence of conformal filling in for some classes of 136 SUN-YUNG ALICE CHANG

Riemannian manifolds. A plausible candidate for the problem to have a positive an- swer is the class of metrics (S 3; h) with the scalar curvature of h being positive. In this case by a result of Marques [2012], the set of such metrics is path-connected, the non- existence argument of Gursky–Han, and Gursky–Han–Stolz (Gursky and Han [2017], Gursky, Han, and Stolz [2018]) also does not apply. One hopes that our compactness argument would lead via either the continuity method or degree theory to the existence of conformal filling in for this class of metrics. We remark some related program for the problem has been outlined in Anderson [1989, 2008]. The first observation is one of the difficulty of the problem is existence of some “non- local” term. To see this, we have the asymptotic behavior of the compactified metric g of CCE manifold (X n+1;M n; g+) with conformal infinity (M n; h) (Graham [2000], Fefferman and Graham [2012]) which in the special case when n = 3 takes the form g := r2g+ = h + g(2)r2 + g(3)r3 + g(4)r4 +



on an asymptotic neighborhood of M (0; ), where r denotes the geodesic defining (2) 1 function of g. It turns out g = 2 Ah and is determined by h (we call such terms (3) local terms), T rhg = 0, while

(3) 1 @ g = (Ricg ) ˛;ˇ 3 @n ˛;ˇ where ˛; ˇ denote the tangential coordinate on M , is a non-local term which is not determined by the boundary metric h. We remark that h together with g(3) determine the asymptotic behavior of g (Fefferman and Graham [ibid.], Biquard [2008]). We now observe that different choices of the defining function r give rise to different conformal metric of hˆ in [h] on M . For convenience, In the rest of this article, we choose the representative hˆ = hY be the Yamabe metric with constant scalar curvature in [h] and denote it by h and its corresponding geodesic defining function by r. Similarly one might ask what is a “good’ representative of gˆ [g] on X? Our first attempt is to 2 choose gˆ := gY , a Yamabe metric in [g]. The difficulty of this choice is that it is not Y Y clear how to control the boundary behavior of g M in terms of h . j We also remark that in seeking the right conditions for the compactness problem, due to the nature of the problem, the natural conditions imposed should be conformally invariant. In the statement of the results below, for a CCE manifold (X 4;M 3; g+), and a con- formal infinity (M; [h]) with the representative h = hY [h], we solve the PDE 2 (5.30) ∆g+ w = 3 2w and denote g = e g be the “Fefferman–Graham” compactification metric with g M =  j h. + We recall that Qg 0, hence the renormalized volume of (X; M; g ) is a multiple  Á of 1 @ Z  (A )dv = 2 I T d = I R d : 2 g g g h g h X  M  6 M @n

Before we state our results, we recall formulas for the specific g metric in a model case. CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 137

4 3 Lemma 5.1. On (B ;S ; gH),

(1 x 2) 2 4 g = e j j dx on B j j 3 Qg 0;Tg 2 on S  Á  Á (3) (g) 0 Á and Z  (A )dv = 8 2: 2 g g B4  We will first state a perturbation result for the compactness problem.

4 3 + 4 Theorem 5.1. Let (B ;S ; gi ) be a family of oriented CCE on B with boundary 3 f f g g S . We assume the boundary Yamabe metric hi in conformal infinity M is of non- negative type. Let g be the corresponding FG compactification. Assume f ig k+3 1. The boundary Yamabe metrics hi form a compact family in C norm with f g k 2; and there exists some positive constant c1 > 0 such that the Yamabe  constant for the conformal infinity [hi ] is bounded uniformly from below by c1, that is,

Y (M; [hi ]) c1;  2. There exists some small positive constant " > 0 such that for all i

Z 2 (5.31) 2(A )dv 8 ": gi gi B4 

Then the family of the g is compact in C k+2;˛ norm for any ˛ (0; 1) up to a diffeo- i 2 morphism fixing the boundary.

Before we sketch the proof of the theorem, we will first mention that on (B4;S 3; g), for a compact metric g with totally geodesic boundary, Gauss–Bonnet–Chern formula takes the form:

2 4 3 2 Z 2 8 (B ;S ) = 8 = ( W g + 2(Ag ))dvg ; B4 jj jj which together with the conformal invariance of the L2 norm of the Weyl tensor, imply that the condition (5.31) in the statement of the Theorem 5.1 is equivalent to

Z 2 (5.32) W + dv + ": g g B4 jj jj i i Ä What is less obvious is that in this setting, we also have other equivalence conditions as stated in Corollary (5.1) below. This is mainly due to following result by Li, Qing, and Shi [2017]. 138 SUN-YUNG ALICE CHANG

Proposition 5.1. Assume that (X n+1; g+) is a CCE manifold with C 3 regularity whose conformal infinity is of positive Yamabe type. Let p X be a fixed point and t > 0. 2 Then n Y (@X; [h]) 2 Vol(@B + (p; t)) Vol(B + (p; t)) Â Ã g g 1 n Y (S ; [gc]) Ä Vol(@BgH (p; t)) Ä Vol(BgH (p; t)) Ä

+ where Bg (p; t) and BgH (p; t) are geodesic balls.

Corollary 5.1. Let X = B4;M = @X = S 3; g+ be a 4-dimensional oriented CCE f g on X with boundary @X. Assume the boundary Yamabe metric h = hY in the conformal 3 infinity of positive type and Y (S ; [h]) > c1 for some fixed c1 > 0 and h is bounded in C k+3 norm with k 5. Let g be the corresponding FG compactification. Then the   following properties are equivalent:

1. There exists some small positive number " > 0 such that

(5.33) Z (A )dv 82 ": g g X  

2. There exists some small positive number " > 0 such that

Z 2 W g+ dvg+ ": X jj jj Ä

3. There exists some small positive number "1 > 0 such that

3 3 3 Y (S ; [gc]) Y (S ; [h]) > Y (S ; [gc]) "1  3 where gc is the standard metric on S .

4. There exists some small positive number "2 > 0 such that for all metrics g with 3 boundary metric h same volume as the standard metric gc on S , we have

T (g) 2 "2: 

5. There exists some small positive number "3 > 0 such that

(3) (g) "3: j j Ä

Where all the "i (i = 1,2,3) tends to zero when " tends to zero and vice versa for each i.

Another consequence of Theorem 5.1 is the “uniqueness” of the Graham–Lee metrics mentioned in section 4a.

Corollary 5.2. There exists some " > 0, such that for all metrics h on S 3 with h 4 3 + jj gc C < ", there exists a unique CCE filling in (B ;S ; g ) of h. jj 1 CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 139

Sketch proof of Theorem 5.1. We refer the readers to the articles Chang and Ge [2018] and Chang, Ge, and Qing [2018], both will soon be posted on arXiv for details of the arguments, here we will present a brief outline. We first state a lemma summarizing some analytic properties of the metrics g: Lemma 5.2. On a CCE manifold (X 4;M 3; g+), where the scalar curvature of the conformal infinity (M; h) is positive. Assume h is at least C l smooth for l 3. Denote 2w +  g = e g the FG compactification. Then (1) Q(g) 0, Á w (2) Rg > 0, which implies in particular g w e 1.  jr  j Ä (3) g is Bach flat and satisfies an -regularity property, which implies in particular, once it is C 3 smooth, it is C l smooth for l 3.  We remark that statement (2) in the Lemma above follows from a continuity argu- ment via some theory of scattering matrix (see Case and Chang [2016a,b]), with the starting point of the argument the positive scalar curvature metric constructed by J. Lee which we have mentioned earlier in Lemma 4.1.

Sketch proof of Theorem 5.1. Proof of the theorem is built on contradiction argu- ments. We first note, assuming the conclusion of the theorem does not hold, then there is a sequence of g which is not compact so that the L2 norm of its Weyl tensor of f ig the sequence tends to zero.

Our main assertion of the proof is that the C 1 norm of the curvature of the family gi remains uniformly bounded. f g 2 Assume the assertion is not true, we rescale the metric g¯i = Ki gi  where there exists some point pi X such that 2 K2 = max sup Rm ; rsup Rm = Rm (p )( or q Rm (p )) i gi  gi  gi  i gi  i f B4 j j B4 jr jg j j jr j

4 We mark the accumulation point pi as 0 B . Thus, we have 2

(5.34) Rmg¯ (0) = 1 or Rmg¯ (0) = 1 j i j jr i j

2w¯i + 2w¯i We denote the corresponding defining function w¯i so that g¯i = e gi ; that is e = 2 2wi ¯ K e and denote hi := g¯i S 3 . We remark that the metrics g¯i also satisfy the condi- i j tions in Lemma 5.2. As 0 B4 is an accumulation point, depending on the location of 0, we call it either 2 an interior or a boundary blow up point; in each case, we need a separate argument but for simplicity here we will assume we have 0 S 3 is a boundary blow up point. In this 2 4 case, we denote the (X ; g ) the Gromov–Hausdorff limit of the sequence (B ; g¯i ), 1 1 and h := g S 3 . 1 1j 140 SUN-YUNG ALICE CHANG

Our first observation is that it follows from the assumption (1) in the statement of the theorem, we have (@X ; h ) = (R3; dx 2). 1 1 j j

Our second assertion is that by the estimates in Lemma 5.2, one can show w¯i con- verges uniformly on compacta on X , we call the limiting function w¯ . Hence, the + 1 2w¯ + 1 + corresponding metric g with g = e 1 g exists, satisfying Ricg+ = 3g , i.e., 1 1 1 + 1 1 the resulting g is again conformal to a Poincare Einstein metric g with Wg+ 0. 1 1 jj 1 jj Á Our third assertion is that (X ; g+ ) is (up to an isometry) the model space 1 1 dx 2 + dy 2 (R4 ; g := j j j j ) + H y2 where R4 = (x; y) R4 y > 0 . We can then apply a Liouville type PDE argument + f 2 j g to conclude w¯ = log y. 1 Thus g is in fact the flat metric dx 2 + dy 2, which contradicts the marking prop- 1 j j j j erty (5.34). This contradiction establishes the main assertion.

Once the C 1 norm of the curvature of the metric g is bounded, we can apply f ig some further blow-up argument to show the diameter of the sequence of metrics is uniformly bounded, and apply a version of the Gromov–Hausdorff compactness result (see Cheeger, Gromov, and Taylor [1982]) for compact manifolds with totally geodesic boundary to prove that g forms a compact family in a suitable C l norm for some f ig l 3. This finishes the proof of the theorem.  We end this discussion by mentioning that, from Theorem 5.1, an obvious question to ask is if we can extend the perturbation result by improving condition (5.31) to the condition RB4 2 > 0. This is a direction we are working on but have not yet be able to accomplish. Below are statements of two more general theorems that we have obtained in (Chang and Ge [2018], Chang, Ge, and Qing [2018]). Theorem 5.2. Under the assumption (1) as in Theorem 5.1, assume further the T cur- vature on the boundary T = 1 @ R satisfies the following condition i 12 @n gi  I (5.35) lim inf inf inf Ti 0: r 0 i S 3  ! @B(x;r) Then the family of metrics g is compact in C k+2;˛ norm for any ˛ (0; 1) up to a f ig 2 diffeomorphism fixing the boundary, provided k 5.  Theorem 5.3. Under the assumption (1) as in Theorem 5.1, assume further that there (3) 3 1 is no concentration of Si := (gi) -tensor defined on S in L norm for the gi metric in the following sense, I (5.36) lim sup sup Si = 0: r 0 x j j ! i @B(x;r) Then, the family of the metrics g is compact in C k+2;˛ norm for any ˛ (0; 1) up f ig 2 to a diffeomorphism fixing the boundary, provided k 2.  CONFORMAL GEOMETRY ON FOUR MANIFOLDS – NOETHER LECTURE 141

The reason we can pass the information from the T curvature in Theorem 5.2 to the S tensor in Theorem 5.3 is due to the fact that for the blow-up limiting metric g , 1 Tg 0 if and only if Sg 0. 1 Á 1 Á It remains to show if there is a connection between condition (5.35) in Theorem 5.2 to the positivity of the renormalized volume, i.e. when RX 2(Ag )dvg > 0.

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