A Paneitz-type operator for CR pluriharmonic functions
aa r X i v : . [ m a t h . DG ] S e p A PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONICFUNCTIONS
JEFFREY S. CASE AND PAUL YANG
Abstract.
We introduce a fourth order CR invariant operator on plurihar-monic functions on a three-dimensional CR manifold, generalizing to the ab-stract setting the operator discovered by Branson, Fontana and Morpurgo. Fora distinguished class of contact forms, all of which have vanishing Hirachi- Q curvature, these operators determine a new scalar invariant with propertiesanalogous to the usual Q -curvature. We discuss how these are similar to the(conformal) Paneitz operator and Q -curvature of a four-manifold, and describeits relation to some problems for three-dimensional CR manifolds. Introduction
It is well-known that there is a deep analogy between the study of three-dimensionalCR manifolds and of four-dimensional conformal manifolds. Two important ingre-dients in the study of the latter are the Paneitz operator P and the Q -curvature Q . Given a metric g , the Paneitz operator is a formally self-adjoint fourth-orderdifferential operator of the form ∆ plus lower-order terms, while the Q -curvatureis a scalar invariant of the form ∆ R plus lower-order terms, where R is the scalarcurvature of g and “order” is measured according to the number of derivatives takenof g . The pair ( P , Q ) generalizes to four-dimensions many important propertiesof the pair ( − ∆ , K ) of the Laplacian and the Gauss curvature of a two-manifold.For example, if ( M , g ) is a Riemannian manifold and ˆ g = e σ g is another choiceof metric, then e σ ˆ P ( f ) = P ( f )(1.1) e σ ˆ Q = Q + P ( σ )(1.2)for all f ∈ C ∞ ( M ). Since also P (1) = 0, the transformation formula (1.2) impliesthat on a compact conformal manifold ( M , [ g ]), the integral of the Q -curvatureis a conformal invariant; indeed, the Gauss–Bonnet–Chern formula states that thisintegral is a linear combination of the Euler characteristic of M and the integralof a pointwise conformal invariant, namely the norm of the Weyl tensor. Thepair ( P , Q ) also appears in the linearization of the Moser–Trudinger inequality.Denoting by ( S , g ) the standard four-sphere with g a metric of constant sectionalcurvature one, it was proven by Beckner [1], and later by Chang and the second Mathematics Subject Classification.
Primary 32V05; Secondary 53C24.
Key words and phrases. pluriharmonic functions, pseudo-Einstein, Paneitz operator, Q -curvature, P -prime operator, Q -prime curvature.JSC was partially supported by NSF Grant No. DMS-1004394.PY was partially supported by NSF Grant No. DMS-1104536. author [9] using a different technique, that(1.3) ˆ S u P u + 2 ˆ S Q u − (cid:18) ˆ S Q (cid:19) log (cid:18) S e u (cid:19) ≥ u ∈ C ∞ ( S ), and that equality holds if and only if e u g is an Einstein metricon ( S , g ).A natural question is whether there exist analogues of P and Q defined fora three-dimensional pseudohermitian manifold ( M , J, θ ). In a certain sense thisis already known; the compatibility operator studied by Graham and Lee [18] is afourth-order CR invariant operator with leading order term ∆ b + T and Hirachi [20]has identified a scalar invariant Q which is related to P through a change of con-tact form in a manner analogous to (1.2). However, while the total Q -curvatureof a compact three-dimensional CR manifold is indeed a CR invariant, it is alwaysequal to zero. Moreover, the Q -curvature of the standard CR three-sphere vanishesidentically; indeed, this is true for the boundary of any strictly pseudoconvex do-main [15], as is explained in Section 4. In particular, while (1.3) is true on the CRthree-sphere, it is trivial, as it only states that the Paneitz operator is nonnegative.Using spectral methods, Branson, Fontana and Morpurgo [3] have recently iden-tified a new operator P ′ on the standard CR three-sphere ( S , J, θ ) such that P ′ is of the form ∆ b plus lower-order terms, P ′ is invariant under the action ofthe CR automorphism group of S , and P ′ appears in an analogue of (1.3) inwhich the exponential term is present. There is, however, a catch: the operator P ′ acts only on the space P of CR pluriharmonic functions on S , namely thosefunctions which are the boundary values of pluriharmonic functions in the ball { ( z, w ) : | z | + | w | < } ⊂ C . The space of CR pluriharmonic functions on S is itself invariant under the action of the CR automorphism group, so it makessense to discuss the invariance of P ′ . Using this operator, Branson, Fontana andMorpurgo [3] showed that(1.4) ˆ S u P ′ u + 2 ˆ S Q ′ u − (cid:18) ˆ S Q ′ (cid:19) log (cid:18) S e u (cid:19) ≥ u ∈ P , where Q ′ = 1 and equality holds in (1.4) if and only if e u θ is atorsion-free contact form with constant Webster scalar curvature.Formally, the operator P ′ is constructed using Branson’s principle of analyticcontinuation in the dimension [2]. More precisely, there exists in general dimensionsa fourth-order CR invariant operator with leading order term ∆ b + T , which weshall also refer to as the Paneitz operator. On the CR spheres, this is an intertwiningoperator, and techniques from representation theory allow one to quickly computethe spectrum of this operator. By carrying out this program, one observes that thePaneitz operator on the standard CR three-sphere kills CR pluriharmonic functions,and moreover, the Paneitz operator P ,n on the standard CR (2 n +1)-sphere acts onCR pluriharmonic functions as n − times a well-defined operator, called P ′ . Oneobservation in [3] is that this operator is in fact a fourth-order differential operatoracting on CR pluriharmonic functions which is, in a suitable sense, CR invariant.The purpose of this article is to show that there is a meaningful definition ofthe “ P ′ -operator” on general three-dimensional CR manifolds enjoying the samealgebraic properties as the operator P ′ defined in [3], and also to investigate thepossibility of defining a scalar invariant Q ′ which is related to P ′ in a manneranalogous to the way in which the Q -curvature is related to the Paneitz operator. PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 3
It turns out that one cannot define Q ′ in a meaningful way for a general choiceof contact form on a CR three-manifold, though one can for a distinguished classof contact forms, namely the so-called pseudo-Einstein contact forms. These areprecisely those contact forms which are locally volume-normalized with respect toa closed section of the canonical bundle, which is a meaningful consideration indimension three (cf. [25] and Section 3). Having made these definitions, we willalso begin to investigate the geometric meaning of these invariants.To describe our results, let us begin by discussing in more detail the ideas whichgive rise to the definitions of P ′ and Q ′ . To define P ′ , we follow the same strategyof Branson, Fontana, and Morpurgo [3]. First, Gover and Graham [16] have shownthat on a general CR manifold ( M n +1 , J ), one can associate to each choice ofcontact form θ a formally-self adjoint real fourth-order operator P ,n which hasleading order term ∆ b + T , and that this operator is CR covariant. On three-dimensional CR manifolds, this reduces to the well-known operator P := P , = ∆ b + T − ∇ α A αβ ∇ β which, through the work of Graham and Lee [18] and Hirachi [20], is known toserve as a good analogue of the Paneitz operator of a four-dimensional conformalmanifold. As pointed out by Graham and Lee [18], the kernel of P (as an operatoron a three-dimensional CR manifold) contains the space P of CR pluriharmonicfunctions, and thus one can ask whether the operator P ′ := lim n → n − P ,n | P is well-defined. As we verify in Section 4, this is the case. It then follows fromstandard arguments (cf. [4]) that if ˆ θ = e σ θ is any other choice of contact form,then the corresponding operator c P ′ is related to P ′ by(1.5) e σ c P ′ ( f ) = P ′ ( f ) + P ( σf )for any f ∈ P . Thus the relation between P ′ and P is analogous to the rela-tion (1.2) between the Q -curvature and the Paneitz operator; more precisely, the P ′ -operator can be regarded as a Q -curvature operator in the sense of Branson andGover [4]. Moreover, since the Paneitz operator is self-adjoint and kills plurihar-monic functions, the transformation formula (1.5) implies that e σ c P ′ ( f ) = P ′ ( f ) mod P ⊥ for any f ∈ P , returning P ′ to the status of a Paneitz-type operator. This is thesense in which the P ′ -operator is CR invariant, and is the way that it is studiedin (1.4).From its construction, one easily sees that P ′ (1) is exactly Hirachi’s Q -curvature.Thus, unlike the Paneitz operator, the P ′ -operator does not necessarily kill con-stants. However, there is a large and natural class of contact forms for which the P ′ -operator does kill constants, namely the pseudo-Einstein contact forms; see Sec-tion 3 for their definition. It turns out that two pseudo-Einstein contact forms ˆ θ and θ must be related by a CR pluriharmonic function, log ˆ θ/θ ∈ P (cf. [25]). If ( M , J )is the boundary of a domain in C , such contact forms exist in profusion, arising assolutions to Fefferman’s Monge-Amp`ere equation (cf. [14, 15]). In this setting, itis natural to ask whether there is a scalar invariant Q ′ such that P ′ (1) = n − Q ′ . JEFFREY S. CASE AND PAUL YANG
This is true; we will show that if ( M , J, θ ) is a pseudo-Einstein manifold, then thescalar invariant Q ′ := lim n → n − P ,n (1)is well-defined. As a consequence, if ˆ θ = e σ θ is another pseudo-Einstein contactform (in particular, σ ∈ P ), then(1.6) e σ c Q ′ = Q ′ + P ′ ( σ ) + 12 P ( σ ) . Taking the point of view that P ′ is a Paneitz-type operator, we may also write e σ c Q ′ = Q ′ + P ′ ( σ ) mod P ⊥ . The upshot is that, on the standard CR three-sphere, Q ′ = 1, so that this indeedrecovers the interpretation of the Beckner–Onofri-type inequality (1.4) of Branson–Fontana–Morpurgo [3] as an estimate involving some sort of Paneitz-type operatorand Q -type curvature. Additionally, we also see from (1.6) that the integral of Q ′ is a CR invariant; more precisely, if ( M , J ) is a compact CR three-manifold and θ, ˆ θ are two pseudo-Einstein contact forms, then ˆ M c Q ′ ˆ θ ∧ d ˆ θ = ˆ M Q ′ θ ∧ dθ. In conformal geometry, the total Q -curvature plays an important role in control-ling the topology of the underlying manifold. For instance, the total Q -curvaturecan be used to prove sphere theorems (e.g. [19, Theorem B] and [8, Theorem A]).We will prove the following CR analogue of Gursky’s theorem [19, Theorem B]. Theorem 1.1.
Let ( M , J, θ ) be a compact three-dimensional pseudo-Einstein man-ifold with nonnegative Paneitz operator and nonnegative CR Yamabe constant.Then ˆ M Q ′ θ ∧ dθ ≤ ˆ S Q ′ θ ∧ dθ , with equality if and only if ( M , J ) is CR equivalent to the standard CR three sphere. Here, the CR Yamabe constant of a CR manifold ( M , J ) is the infimum of thetotal Webster scalar curvature over all contact forms θ such that ´ θ ∧ dθ = 1 (cf.[22]). The proof of Theorem 1.1 relies upon the existence of a CR Yamabe contactform — that is, the existence of a smooth unit-volume contact form with constantWebster scalar curvature equal to the CR Yamabe constant [12, 22]. In particular,it relies on the CR Positive Mass Theorem [12]. One complication which does notarise in the conformal case [19] is the possibility that the CR Yamabe contact formmay not be pseudo-Einstein. We overcome this difficulty by computing how thelocal formula (4.6) for Q ′ transforms with a general change of contact form; i.e.without imposing the pseudo-Einstein assumption. For details, see Section 6.In conformal geometry, the total Q -curvature also arises when considering theEuler characteristic of the underlying manifold. Burns and Epstein [5] have shownthat there is a biholomorphic invariant, now known as the Burns–Epstein invariant,of the boundary of a strictly pseudoconvex domain which is related to the Eulercharacteristic of the domain in a similar way. It turns out that the Burns–Epsteininvariant is a constant multiple of the total Q ′ -curvature, and thus there is a nicerelationship between the total Q ′ -curvature and the Euler characteristic. PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 5
Theorem 1.2.
Let ( M , J ) be a compact CR manifold which admits a pseudo-Einstein contact form θ , and denote by µ ( M ) the Burns–Epstein invariant of ( M , J ) . Then µ ( M ) = − π ˆ M Q ′ θ ∧ dθ. In particular, if ( M , J ) is the boundary of a strictly pseudoconvex domain X , then ˆ X (cid:18) c − c (cid:19) = χ ( X ) − π ˆ M Q ′ θ ∧ dθ, where c and c are the first and second Chern forms of the K¨ahler–Einstein metricin X obtained by solving Fefferman’s equation and χ ( X ) is the Euler characteristicof X . While we were discussing a preliminary version of this work at Banff in Summer2012, it was suggested to us by Kengo Hirachi that a version of Theorem 1.2 shouldbe true. It was then pointed out to us by Jih-Hsin Cheng that Theorem 1.2 can beproved by using the formula given by Burns and Epstein [5] (see also [11]) for theirinvariant. This fact has since been independently verified by Hirachi [21], to whichwe refer the reader for the details of the verification of Theorem 1.2.Finally, we point out that much of the background described above generalizesto higher dimensions. On any even-dimensional Riemannian manifold ( M n , g )there exists a pair ( P n , Q n ) of a conformally-invariant differential operator P n of the form ( − ∆) n plus lower order terms, the so-called GJMS operators [17], andscalar invariants Q n of the form ( − ∆) n − R plus lower-order terms, the so-called(critical) Q -curvatures [2], which satisfy transformation rules analogous to (1.1)and (1.2). On the standard 2 n -sphere, Beckner [1] and Chang–Yang [9] showed thatthe analogue of (1.3) still holds, including the characterization of equality. Likewise,Branson, Fontana and Morpurgo [3] defined operators P ′ n +2 on the standard CR(2 n + 1)-sphere which are CR invariant operators of order 2 n + 2 and for whichan analogue of (1.4) holds, including the characterization of equality, where again Q ′ n +2 are only identified as explicit constants. After a preliminary version ofthis article was presented at Banff in Summer 2012, Hirachi [21] showed how touse the ambient calculus to extend the P ′ -curvature and Q ′ -curvature to higherdimensions in such a way that the transformation formulae (1.5) and (1.6) hold. In aforthcoming work with Rod Gover, we produce tractor formulae for the P ′ -operatorand the Q ′ -curvature. This allows us to produce for pseudo-Einstein manifolds withvanishing torsion a product formula for the P ′ -operator and an explicit formula forthe Q ′ -curvature, giving a geometric derivation of the formulae given by Branson,Fontana and Morpurgo [3].This article is organized as follows. In Section 2, we recall some basic definitionsand facts in CR geometry, and in particular recall the depth of the analogy betweenaspects of conformal and CR geometry. In Section 3, we introduce the notion ofa pseudo-Einstein contact form on a three-dimensional CR manifold, and exploresome basic properties of such forms. In Section 4, we give a general formula for thePaneitz operator on a CR manifold ( M n +1 , J, θ ). We then use this formula to givethe definitions of the P ′ -operator and the Q ′ -curvature, and establish some of theirbasic properties. In Section 5, we check by direct computation that the P ′ -operatorsatisfies the correct transformation law. Indeed, this computation shows that P ′ nolonger satisfies this rule if it is considered on a space strictly larger than the space JEFFREY S. CASE AND PAUL YANG of CR pluriharmonic functions. In Section 6, we check by direct computation thatthe Q ′ -curvature satisfies the correct transformation law, and use this computationto prove Theorem 1.1. In the appendices, we will derive in two different ways thelocal formula for the CR Paneitz operator in general dimension. First, Appendix Agives the derivation using the CR tractor calculus [16]. Second, Appendix B givesthe derivation using Lee’s construction [24] of the Fefferman bundle.2. CR Geometry
Throughout this article, we will follow the conventions used by Gover and Gra-ham [16] for describing CR and pseudohermitian invariants and performing localcomputations using a choice of contact form. These conventions are identical to thethe conventions used by Lee in his work on pseudo-Einstein structures [25], exceptthat we will sometimes describe invariants as densities rather than functions. Thishas the effect that exponential factors will generally not appear in our formulaefor how these invariants transform under a change of contact form. Both for theconvenience of the reader and to hopefully avoid any confusion caused by the manydifferent notations used in the literature, we use this section to make precise theseconventions as necessary for this article.2.1.
CR and pseudohermitian manifolds. A CR manifold is a pair ( M n +1 , J )of a smooth oriented (real) (2 n + 1)-dimensional manifold together with a formallyintegrable complex structure J : H → H on a maximally nonintegrable codimensionone subbundle H ⊂ T M . In particular, the bundle E = H ⊥ ⊂ T ∗ M is orientableand any nonvanishing section θ of E is a contact form ; i.e. θ ∧ ( dθ ) n is nonvanishing.We will assume further that ( M n +1 , J ) is strictly pseudoconvex , meaning thatthe symmetric tensor dθ ( · , J · ) on H ∗ ⊗ H ∗ is positive definite; since E is one-dimensional, this is independent of the choice of contact form θ .Given a CR manifold ( M n +1 , J ), we can define the subbundle T , of the com-plexified tangent bundle T C M as the + i -eigenspace of J , and T , as its conjugate.We likewise denote by Λ , the space of (1 , T ∗ C M which annihilates T , — and by Λ , its conjugate. The canonical bundle K is the complex line-bundle K = Λ n +1 (cid:0) Λ , (cid:1) .A pseudohermitian manifold is a triple ( M n +1 , J, θ ) of a CR manifold ( M n +1 , J )together with a choice of contact form θ . The assumption that dθ ( · , J · ) is positivedefinite implies that the Levi form L θ ( U ∧ ¯ V ) = − idθ ( U ∧ ¯ V ) defined on T , is apositive-definite Hermitian form. Since another choice of contact form ˆ θ is equiva-lent to a choice of (real-valued) function σ ∈ C ∞ ( M ) such that ˆ θ = e σ θ , and theLevi forms of ˆ θ and θ are related by L ˆ θ = e σ L θ , we see that the analogy betweenCR geometry and conformal geometry begins through the similarity of choosing acontact form or a metric in a conformal class (cf. [22]).Given a pseudohermitian manifold ( M n +1 , J, θ ), the Reeb vector field T is theunique vector field such that θ ( T ) = 1 and T y dθ = 0. An admissible coframe isa set of (1 , { θ α } nα =1 whose restriction to T , forms a basis for (cid:0) T , (cid:1) ∗ and such that θ α ( T ) = 0 for all α . Denote by θ ¯ α = θ α the conjugate of θ α . Then dθ = ih α ¯ β θ α ∧ θ ¯ β for some positive definite Hermitian matrix h α ¯ β . Denote by { T, Z α , Z ¯ α } the frame for T C M dual to { θ, θ α , θ ¯ α } , so that the Levi form is L θ (cid:0) U α Z α , V ¯ α Z ¯ α (cid:1) = h α ¯ β U α V ¯ β . PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 7
Tanaka [26] and Webster [27] have defined a canonical connection on a pseu-dohermitian manifold ( M n +1 , J, θ ) as follows: Given an admissible coframe { θ α } ,define the connection forms ω αβ and the torsion form τ α = A αβ θ β by the relations dθ β = θ α ∧ ω αβ + θ ∧ τ β ,ω α ¯ β + ω ¯ βα = dh α ¯ β ,A αβ = A βα , where we use the metric h α ¯ β to raise and lower indices; e.g. ω α ¯ β = h γ ¯ β ω αγ . Inparticular, the connection forms are pure imaginary. The connection forms define the pseudohermitian connection on T , by ∇ Z α = ω αβ ⊗ Z β , which is the uniqueconnection preserving T , , T , and the Levi form.The curvature form Π αβ := dω αβ − ω αγ ∧ ω γβ can be writtenΠ αβ = R αβγ ¯ δ θ γ ∧ θ ¯ δ mod θ, defining the curvature of M . The pseudohermitian Ricci tensor is the contraction R α ¯ β := R γγα ¯ β and the pseudohermitian scalar curvature is the contraction R := R αα . As shown by Webster [27], the contraction Π γγ is given by(2.1) Π γγ = dω γγ = R α ¯ β θ α ∧ θ ¯ β + ∇ β A αβ θ α ∧ θ − ∇ ¯ β A ¯ α ¯ β θ ¯ α ∧ θ. For computational and notational efficiency, it will usually be more useful towork with the pseudohermitian Schouten tensor P α ¯ β := 1 n + 2 (cid:18) R α ¯ β − n + 1) Rh α ¯ β (cid:19) and its trace P := P αα = R n +1) . The following higher order derivatives T α = 1 n + 2 (cid:0) ∇ α P − i ∇ β A αβ (cid:1) S = − n (cid:16) ∇ α T α + ∇ ¯ α T ¯ α + P α ¯ β P α ¯ β − A αβ A αβ (cid:17) will also appear frequently (cf. [16, 24]).In performing computations, we will usually use abstract index notation, so forexample τ α will denote a (1 , ∇ α ∇ β f will denote the (2 , Lemma 2.1.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold. Then ∇ α ∇ β f − ∇ β ∇ α f = 0 , ∇ ¯ β ∇ α f − ∇ α ∇ ¯ β f = ih α ¯ β ∇ f, ∇ α ∇ f − ∇ ∇ α f = A αγ ∇ γ f, ∇ β ∇ τ α − ∇ ∇ β τ α = A γβ ∇ γ τ α + τ γ ∇ α A γβ , where ∇ denotes the derivative in the direction T . The following consequences of the Bianchi identities established in [25, Lemma 2.2]will also be useful.
Lemma 2.2.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold. Then ∇ α P α ¯ β = ∇ ¯ β P + ( n − T ¯ β (2.2) ∇ R = ∇ α ∇ β A αβ + ∇ α ∇ β A αβ . (2.3) JEFFREY S. CASE AND PAUL YANG
In particular, combining the results of Lemma 2.1 and Lemma 2.2 yields(2.4) ∆ b R − n Im ∇ α ∇ β A αβ = − ∇ α (cid:0) ∇ α R − in ∇ β A αβ (cid:1) . An important operator in the study of pseudohermitian manifolds is the sub-laplacian ∆ b := − ( ∇ α ∇ α + ∇ α ∇ α ) . Defining the subgradient ∇ b u as the projection of du onto H ∗ ⊗ C — that is, ∇ b f = ∇ α f + ∇ ¯ α f — it is easy to show that ˆ M u ∆ b v θ ∧ dθ n = ˆ M h∇ b u, ∇ b v i θ ∧ dθ n for any u, v ∈ C ∞ ( M ), at least one of which is compactly supported, and where h· , ·i denotes the Levi form.One important consequence of Lemma 2.1 is that the operator C has the followingtwo equivalent forms: Cf := ∆ b f + n ∇ f − in ∇ β (cid:0) A αβ ∇ α f (cid:1) + 2 in ∇ β ( A αβ ∇ α f )= 4 ∇ α (cid:0) ∇ α ∇ β ∇ β f + inA αβ ∇ β f (cid:1) . (2.5)In dimension n = 1, the operator C is the compatibility operator found by Grahamand Lee [18]. Hirachi [20] later observed that in this dimension C is a CR covariantoperator, in the sense that it satisfies a particularly simple transformation formulaunder a change of contact form. Thus, in this dimension C is the CR Paneitzoperator P ; for further discussion, see Section 4.2.2. CR pluriharmonic functions.
Given a CR manifold ( M n +1 , J ), a CRpluriharmonic function is a function u ∈ C ∞ ( M ) which is locally the real partof a CR function v ∈ C ∞ ( M ; C ); i.e. u = Re( v ) for v satisfying ∂v := ∇ ¯ α v = 0.We will denote by P the space of pluriharmonic functions on M , which is usu-ally an infinite-dimensional vector space. When additionally a choice of contactform θ is given, Lee [25] proved the following alternative characterization of CRpluriharmonic functions which does not require solving ∂v = 0. Proposition 2.3.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold. A function u ∈ C ∞ ( M ) is CR pluriharmonic if and only if B α ¯ β u := ∇ ¯ β ∇ α u − n ∇ γ ∇ γ u h α ¯ β = 0 , if n ≥ P α u := ∇ α ∇ β ∇ β u + inA αβ ∇ β u = 0 , if n = 1 . Using Lemma 2.1, it is straightforward to check that (cf. [18])(2.6) ∇ ¯ β (cid:0) B α ¯ β u (cid:1) = n − n P α u. In particular, we see that the vanishing of B α ¯ β u implies the vanishing of P α u when n >
1. Moreover, the condition B α ¯ β u = 0 is vacuous when n = 1, and by (2.6), wecan consider the condition P α u = 0 from Proposition 2.3 as the “residue” of thecondition B α ¯ β u = 0 (cf. Section 3 and Section 4).Note also that, using the second expression in (2.5), we have that C = 4 ∇ α P α . Inparticular, it follows that P ⊂ ker P for three-dimensional CR manifolds ( M , J ).It is easy to see that this is an equality when ( M , J ) admits a torsion-free contactform (cf. [18]), but a good characterization of when equality holds is not yet known. PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 9
CR density bundles.
One generally wants to study CR geometry using
CRinvariants ; i.e. using invariants of a CR manifold ( M n +1 , J ). However, it is fre-quently easier to do geometry by making a choice of contact form θ so as to makeuse of the Levi form and the associated pseudohermitian connection. If one takesthis point of view, it then becomes important to know how the pseudohermitianconnection and the pseudohermitian curvatures transform under a change of con-tact form, and also to have a convenient way to describe objects which transformin a simple way with a change of contact form. This goal is met using CR densitybundles.Given a CR manifold ( M n +1 , J ), choose a ( n + 2)-nd root of the canonicalbundle K and denote it by E (1 , w, w ′ ∈ R with w − w ′ ∈ Z , the ( w, w ′ ) -density bundle E ( w, w ′ ) is the complex linebundle E ( w, w ′ ) = E (1 , ⊗ w ⊗ E (1 , ⊗ w ′ . For our purposes, the important property of E ( w, w ′ ) is that a choice of contactform θ induces an isomorphism between the space E ( w, w ′ ) of smooth sections of E ( w, w ′ ) and C ∞ ( M ; C ), E ( w, w ′ ) ∋ u ∼ = u θ ∈ C ∞ ( M ; C ) , with the property that if ˆ θ = e σ θ is another choice of contact form, then u ˆ θ isrelated to u θ by(2.7) u ˆ θ = e w σ e w ′ ¯ σ u θ ;for details, see [16]. We will also consider density-valued tensor bundles; for exam-ple, we will denote by E α ( w, w ′ ) and E ¯ α ( w, w ′ ) the tensor products T , ⊗ E ( w, w ′ )and T , ⊗ E ( w, w ′ ), respectively, and by E α ( w, w ′ ) and E ¯ α ( w, w ′ ) their respectivespaces of smooth sections. In this way, we may regard the Levi form as the density h α ¯ β ∈ E α ¯ β (1 , w, w )-density bundles and tensor products thereof.In particular, if u ∈ E ( w, w ) is real-valued and we restrict ourselves to real-valuedcontact forms, the transformation rule (2.7) becomes u ˆ θ = e wσ u θ .In [25] (see also [16]), the transformation formulae for the pseudohermitian con-nection and its torsion and curvatures under a change of contact form are given,which we record below: Lemma 2.4.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold and regard the tor-sion A αβ ∈ E αβ (0 , , the pseudohermitian Schouten tensor P α ¯ β ∈ E α ¯ β (0 , , andits trace P = P αα ∈ E ( − , − . Additionally, let f ∈ E ( w, w ) and τ α ∈ E α ( w, w ) . If ˆ θ = e σ θ is another choice of contact form and ˆ A αβ , ˆ P α ¯ β , ˆ P are its torsion, pseu-dohermitian Schouten tensor, and its trace, respectively, then ˆ A αβ = A αβ + i ∇ β ∇ α σ − i ( ∇ α σ )( ∇ β σ )ˆ P α ¯ β = P α ¯ β − (cid:0) ∇ ¯ β ∇ α σ + ∇ α ∇ ¯ β σ (cid:1) − |∇ γ σ | h α ¯ β ˆ P = P + 12 ∆ b σ − n |∇ γ σ | ˆ ∇ α f = ∇ α f + wf ∇ α σ ˆ ∇ f = ∇ f + i ( ∇ α σ )( ∇ α f ) − i ( ∇ α σ )( ∇ α f ) + wf ∇ σ ˆ ∇ α τ β = ∇ α τ β + ( w − τ β ∇ α σ − τ α ∇ β σ ˆ ∇ ¯ β τ α = ∇ ¯ β τ α + wτ α ∇ ¯ β σ + τ γ ∇ γ σh α ¯ β . There are a few technical comments necessary to properly interpret Lemma 2.4.First, we define the norm |∇ γ σ | := ( ∇ γ σ )( ∇ γ σ ). In particular, |∇ γ σ | = h∇ b σ, ∇ b σ i . We define norms on all (density-valued) tensors in a similar way; forexample, | A αβ | = A αβ A αβ and | P α ¯ β | = P α ¯ β P α ¯ β .Second, for these formulae to be valid component-wise, one also needs to changethe admissible frame in which one computes the components of the torsion and CRSchouten tensor. Explicitly, if { θ, θ α , θ ¯ α } is an admissible coframe for the contactform θ , one defines ˆ θ α = θ α + i ( ∇ α σ ) θ and ˆ θ ¯ α by conjugation, ensuring that { ˆ θ, ˆ θ α , ˆ θ ¯ α } is an admissible coframe for thecontact form ˆ θ . In the above formulae, this frame is used to compute the compo-nents of ˆ ∇ α and ˆ ∇ ¯ α , while the coframe { θ, θ α , θ ¯ α } is used to compute the compo-nents of ∇ α and ∇ ¯ α .Third, to regard P ∈ E ( − , −
1) means to extend the function P to a density ρ ∈ E ( − , −
1) by requiring ρ θ = P , and we use h α ¯ β ∈ E α ¯ β (1 ,
1) to raise and lowerindices. This has the effect that, at the level of functions, Lemma 2.4 states thatˆ P = e − σ (cid:18) P + 12 ∆ b σ − n |∇ γ σ | (cid:19) , which is the transformation formula proven in [25]. It also means that we canquickly compute how ∇ α P transforms under a change of contact form: UsingLemma 2.4 with P ∈ E ( − , − c ∇ α b P = ∇ α (cid:18) P + 12 ∆ b σ − n |∇ γ σ | (cid:19) − (cid:18) P + 12 ∆ b σ − n |∇ γ σ | (cid:19) ∇ α σ. These conventions will be exploited heavily in Section 5.3.
Pseudo-Einstein contact forms in three dimensions
In [25], Lee defined pseudo-Einstein manifolds as pseudohermitian manifolds( M n +1 , J, θ ) such that P α ¯ β − n P h α ¯ β = 0 and studied their existence when n ≥ θ is pseudo-Einstein if and only if it islocally volume-normalized with respect to a closed nonvanishing section of K ; thatis, using the terminology of Fefferman and Hirachi [15], θ is pseudo-Einstein if andonly if it is an invariant contact form. While Lee’s definition of pseudo-Einsteincontact forms is vacuous in dimension three, the notion of an invariant contact form PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 11 is not. It turns out that, analogous to Proposition 2.3, there is a meaningful wayto extend the notion of pseudo-Einstein contact forms to the case n = 1 as a higherorder condition on θ which retains the equivalence with invariant contact forms.As this notion will be essential to our discussion of the Q ′ -curvature, and becauseit did not appear elsewhere in the literature at the time the work of this paper wasbeing completed, we devote this section to explaining this three-dimensional notionof pseudo-Einstein contact forms. Definition 3.1.
A pseudohermitian manifold ( M n +1 , J, θ ) is said to be pseudo-Einstein if R α ¯ β − n Rh α ¯ β = 0 , if n ≥ , ∇ α R − i ∇ β A αβ = 0 , if n = 1 . One way to regard this definition is as an analogue of Lee’s characterization [25]of CR pluriharmonic functions from Proposition 2.3. Indeed, a straightforwardcomputation using Lemma 2.2 shows that(3.1) ∇ ¯ β (cid:18) R α ¯ β − n Rh α ¯ β (cid:19) = n − n (cid:0) ∇ α R − in ∇ β A αβ (cid:1) holds for any CR manifold ( M n +1 , J, θ ). In particular, our definition of a three-dimensional pseudo-Einstein manifold can be regarded as the “residue” of the usualdefinition when n ≥ n = 1. To see this, let us first recall what it means for a contactform to be volume-normalized with respect to a section of K . Definition 3.2.
Given a CR manifold ( M n +1 , J ) and a nonvanishing section ω of the canonical bundle K , we say that a contact form θ is volume-normalized withrespect to ω if θ ∧ ( dθ ) n = i n n ! θ ∧ ( T y ω ) ∧ ( T y ω ) . By considering all the terms in dω αα , Lee’s argument [25, Theorem 4.2] establish-ing the equivalence between pseudo-Einstein contact forms and invariant contactforms can be extended to the case n = 1. Theorem 3.3.
Let ( M , J ) be a three-dimensional CR manifold. A contact form θ on M is pseudo-Einstein if and only if for each point p ∈ M , there exists aneighborhood of p in which there is a closed section of the canonical bundle withrespect to which θ is volume-normalized. The main step in the proof of Theorem 3.3 is the following analogue of [25,Lemma 4.1].
Lemma 3.4.
Let ( M , J ) be a three-dimensional CR manifold. A contact form θ on M is pseudo-Einstein if and only if with respect to any admissible coframe { θ, θ α , θ ¯ α } the one-form ω αα + iRθ is closed.Proof. Using (2.1) and the assumption n = 1, it holds in general that(3.2) dω αα = Rh α ¯ β θ α ∧ θ ¯ β + ∇ β A αβ θ α ∧ θ − ∇ ¯ β A ¯ α ¯ β θ ¯ α ∧ θ. It thus follows that d ( ω αα + iRθ ) = 2 i Re (cid:0) ( ∇ α R − i ∇ β A αβ ) θ α ∧ θ (cid:1) , from which the conclusion follows immediately. (cid:3) Proof of Theorem 3.3.
First suppose that θ is volume-normalized with respect to aclosed section ξ ∈ K on a neighborhood U of p , and choose an admissible coframe { θ, θ α , θ ¯ α } such that dθ = iθ α ∧ θ ¯ α . Since ξ ∈ K , there is a function λ ∈ C ∞ ( M, C )such that ξ = λθ ∧ θ α . On the other hand, since θ is volume-normalized with respectto ξ , it must hold that | λ | = 1. Thus, upon replacing θ α by λ − θ α , we have that ξ = θ ∧ θ α .Now, using the definition of the connection one-form ω αα , it holds in generalthat(3.3) dξ = − ω αα ∧ ξ. Since ξ is closed, this shows that ω αα is a (1 , ω αα is also pure imagi-nary, hence ω αα = iuθ for some u ∈ C ∞ ( M ). Differentiating, we see that dω αα = − uθ α ∧ θ ¯ α + i ∇ α uθ α ∧ θ + i ∇ ¯ α uθ ¯ α ∧ θ. It thus follows from (3.2) that R = − u and ∇ β A αβ = i ∇ α u . In particular, θ ispseudo-Einstein.Conversely, suppose that θ is pseudo-Einstein. In a neighborhood of p ∈ M ,let { θ, θ α , θ ¯ α } be an admissible coframe such that dθ = iθ α ∧ θ ¯ α , and define ξ = θ ∧ θ α ∈ K . By (3.3) it holds that dξ = − ω αα ∧ ξ , while by Lemma 3.4 thereexists a function φ such that ω αα + iRθ = idφ. Since ω αα is pure imaginary, we can take φ to be real, whence d (cid:0) e iφ ξ (cid:1) = 0. Since θ is volume-normalized with respect to e iφ ξ , this gives the desired section of K . (cid:3) Another nice property of pseudo-Einstein contact forms when n ≥ n = 1, which is crucial to making sense of the Q ′ -curvature. To see this, let ( M , J, θ ) be a pseudohermitian manifold and define the(1 , W α by W α := ∇ α R − i ∇ β A αβ . Observe that W α vanishes if and only if θ is pseudo-Einstein. As first observed byHirachi [20], W α satisfies a simple transformation formula; given another contactform ˆ θ = e σ θ , a straightforward computation using Lemma 2.4 shows that(3.4) ˆ W α = W α − P α σ, where here we regard W α ∈ E α ( − , − Proposition 3.5.
Let ( M , J, θ ) be a pseudo-Einstein three-manifold. Then theset of pseudo-Einstein contact forms on ( M , J ) is given by { e u θ : u is a CR pluriharmonic function } . Following [25], there are topological obstructions to the existence of an invariantcontact form θ on a three-dimensional CR manifold ( M , J ). However, if ( M , J )is the boundary of a strictly pseudoconvex domain in C , then there always existsa closed section of K , and hence a pseudo-Einstein contact form. This is a slight PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 13 refinement of the observation by Fefferman and Hirachi [15] that any such CRmanifold admits a contact form θ such that the Q -curvature(3.5) Q := − ∇ α W α vanishes. 4. The CR Paneitz and Q -Curvature Operators The CR Paneitz operator in dimension three is well-known and given by P = C .However, in higher dimensions the operator C is not CR covariant. The correctdefinition, in that P is CR covariant, is as follows. Definition 4.1.
Let ( M n +1 , J, θ ) be a CR manifold. The CR Paneitz operator P is the operator P f := ∆ b f + ∇ f − (cid:0) ∇ α ( A αβ ∇ β f ) (cid:1) + 4 Re (cid:18) ∇ ¯ β ( ◦ P α ¯ β ∇ α f ) (cid:19) − n − n Re (cid:0) ∇ β ( P ∇ β f ) (cid:1) + n − Q f. where Q = 2( n + 1) n ( n + 2) ∆ b P − n ( n + 2) Im (cid:0) ∇ α ∇ β A αβ (cid:1) − n − n | A αβ | − n + 1) n | ◦ P α ¯ β | + 2( n − n + 1) n P and ◦ P α ¯ β = P α ¯ β − Pn h α ¯ β is the tracefree part of the CR Schouten tensor.The above expression for the CR Paneitz operator in general dimension does notseem to appear anywhere in the literature, though its existence and two differentmethods to derive the formula have been established by Gover and Graham [16].In particular, their construction immediately implies that the CR Paneitz operatoris CR covariant,(4.1) P : E (cid:18) − n − , − n − (cid:19) → E (cid:18) − n + 32 , − n + 32 (cid:19) . By inspection, it is clear that P is a real, (formally) self-adjoint fourth order oper-ator of the form ∆ b + T plus lower order terms, and thus has the form one expectsof a “Paneitz operator” (cf. [16]). For convenience, we derive in the appendices theabove expression for the CR Paneitz operator using both methods described in [16],namely the CR tractor calculus and restriction from the Fefferman bundle.As mentioned in Section 2, in the critical case n = 1 we have that P ⊂ ker P .Motivated by [3, 4], we define the P ′ -operator corresponding to the CR Paneitz op-erator as a renormalization of the part of P which doesn’t annihilate pluriharmonicfunctions. Definition 4.2.
Let ( M n +1 , J, θ ) be a CR manifold. The P ′ -operator P ′ : P → C ∞ ( M ) is defined by P ′ f = 2 n − P f. When n = 1, we define P ′ by the formal limit P ′ f = lim n → n − P f. The key property of the P ′ -operator, which we check explicitly below, is that theexpression for P ′ as defined in Definition 4.2 is rational in the dimension and doesnot have a pole at n = 1; in particular, it is meaningful to discuss the P ′ -operatoron three-dimensional CR manifolds. Lemma 4.3.
Let ( M n +1 , J, θ ) be a CR manifold. Then the P ′ -operator is givenby P ′ f = 2( n + 1) n ∆ b f − n Im (cid:0) ∇ α ( A αβ ∇ β f ) (cid:1) − n + 1) n Re ( ∇ α ( P ∇ α f ))+ 16( n + 1) n ( n + 2) Re (cid:18) ( ∇ α P − in n + 1) ∇ β A αβ ) ∇ α f (cid:19) + (cid:20) n + 1) n ( n + 2) ∆ b P − n ( n + 2) Im (cid:0) ∇ α ∇ β A αβ (cid:1) − n − n | A αβ | − n + 1) n | ◦ P α ¯ β | + 2( n − n + 1) n P (cid:21) f. (4.2) In particular, if n = 1 , the critical P ′ -operator is given by P ′ f = 4∆ b f − (cid:0) ∇ α ( A αβ ∇ β f ) (cid:1) − ∇ α ( R ∇ α f ))+ 83 Re (cid:0) ( ∇ α R − i ∇ β A αβ ) ∇ α f (cid:1) + 23 (cid:18) ∆ b R −
12 Im ∇ α ∇ β A αβ (cid:19) f. (4.3) Proof.
When n >
1, this follows directly from the definition of the CR Paneitzoperator and the fact that f ∈ P if and only if ∇ ¯ β ∇ α f = µ h α ¯ β for some µ ∈ C ∞ ( M ), which in turn implies, using (2.5), that∆ b f + n ∇ f − n Im (cid:0) ∇ α ( A αβ ∇ β f ) (cid:1) = 0 . Letting n → n = 1. (cid:3) Note that, as an operator on C ∞ ( M ), the P ′ -operator is only determined uniquelyup to the addition of operators which annihilate P . We have chosen the expres-sion (4.3) so that our expression does not involve T -derivatives. In particular, thisallows us to readily connect the P ′ -operator to similar objects already appearingin the literature.(1) On a general CR manifold ( M , J, θ ), P ′ (1) = 23 (cid:0) ∆ b R − ∇ α ∇ β A αβ (cid:1) , which is, using (2.4), Hirachi’s Q -curvature (3.5).(2) On ( S , J, θ ) with its standard CR structure, the P ′ -operator is given by P ′ = 4∆ b + 2∆ b , which is the operator introduced by Branson, Fontana and Morpurgo [3]. PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 15
The means by which we defined the P ′ -operator, and which we will furtheremploy to establish its CR covariance, is called “analytic continuation in the di-mension” (cf. [2]). However, due to the relatively simple form of the expression for P ′ , we can also check its CR covariance directly, as is carried out in Section 5.In the case that ( M , J, θ ) is a pseudo-Einstein manifold, the P ′ -operator takesthe simple form(4.4) P ′ = 4∆ b − ∇ α (cid:0) A αβ ∇ β (cid:1) − ∇ α ( R ∇ α ) . In particular, we see that P ′ annihilates constants, leading us to consider the “ Q -curvature of the P ′ -operator,” which we shall simply call the Q ′ -curvature. Definition 4.4.
Let ( M n +1 , J, θ ) be a pseudo-Einstein manifold. The Q ′ -curvature Q ′ ∈ C ∞ ( M ) is the local invariant defined by Q ′ = 2 n − P ′ (1) = 4( n − P (1) . When n = 1, we define Q ′ as the formal limit Q ′ = lim n → n − P (1) . Again, it is straightforward to give an explicit formula for Q ′ . Lemma 4.5.
Let ( M n +1 , J, θ ) be a pseudo-Einstein manifold. Then the Q ′ -curvature is given by (4.5) Q ′ = 2 n ∆ b R − n | A αβ | + 1 n R . In particular, when n = 1 the Q ′ -curvature is (4.6) Q ′ = 2∆ b R − | A αβ | + R . Proof.
When n >
1, it follows from (3.1) and the pseudo-Einstein assumption that | ◦ P α ¯ β | = 0 and ∆ b R − n Im (cid:0) ∇ α ∇ β A αβ (cid:1) = 0 . Plugging in to (4.2), we see that P ′ (1) = n − n ∆ b R − n − n | A αβ | + ( n − n R . Multiplying by n − then yields the desired result. (cid:3) Let us now verify some basic properties of the P ′ -operator and the Q ′ -curvature.These objects are best behaved and the most interesting in the critical dimension n = 1, so we shall make our statements only in this dimension. Proposition 4.6.
Let ( M , J, θ ) be a pseudohermitian manifold with P ′ -operator P ′ : P → C ∞ ( M ) . Then the following properties hold. (1) P ′ is formally self-adjoint. (2) Given another choice of contact form ˆ θ = e σ θ with σ ∈ C ∞ ( M ) , it holdsthat (4.7) e σ ˆ P ′ ( f ) = P ′ ( f ) + P ( f σ ) for all f ∈ P , where ˆ P ′ denotes the P ′ -operator defined in terms of ˆ θ . Proof.
On a general pseudohermitian manifold ( M n +1 , J, θ ), it follows from Defi-nition 4.2 and the self-adjointness of P that, given u, v ∈ P , n − ˆ M u P ′ v = ˆ M u P v = ˆ M v P u = n − ˆ M v P ′ u, establishing the self-adjointness of P ′ . Likewise, the covariance (4.1) of the CRPaneitz operator implies that for all u ∈ P , n − e n +32 σ ˆ P ′ ( u ) = P (cid:16) e n − σ u (cid:17) = n − P ′ ( u ) + P (cid:16)(cid:16) e n − σ − (cid:17) u (cid:17) . Multiplying both sides by n − and taking the limit n → (cid:3) Remark . It would be nice to have a formula for the critical P ′ -operator which ismanifestly formally self-adjoint on all functions. At present, we have not been ableto find such a formula without the assumption that θ is a pseudo-Einstein contactform, in which case (4.4) gives such a formula.Using the same argument with Q ′ in place of P ′ and P ′ in place of P , we geta similar result for transformation law of the Q ′ -curvature when the contact formis changed by a CR pluriharmonic function. Proposition 4.8.
Let ( M , J, θ ) be a pseudo-Einstein manifold. Given σ ∈ P ,denote ˆ θ = e σ θ . Then (4.8) e σ ˆ Q ′ = Q ′ + P ′ ( σ ) + 12 P ( σ ) . In particular, if M is compact then (4.9) ˆ M ˆ Q ′ ˆ θ ∧ d ˆ θ = ˆ M Q ′ θ ∧ dθ. Proof.
For n > σ ∈ P , we have that (cid:18) n − (cid:19) e n +32 σ ˆ Q ′ = (cid:18) n − (cid:19) Q ′ + P (cid:16) e n − σ − (cid:17) = (cid:18) n − (cid:19) Q ′ + n − P (cid:18) σ + n − σ + O (cid:0) ( n − (cid:1)(cid:19) . Multiplying by n − and taking the limit n → P ′ and P on their respective domains andthe facts that P (1) = 0 for any contact form and P ′ (1) = 0 for any pseudo-Einsteincontact form. (cid:3) We conclude this section with a useful observation about the sign of the P ′ -operator, which can be regarded as a CR analogue of a result of Gursky [19] forthe Paneitz operator in conformal geometry. Proposition 4.9.
Let ( M , J ) be a compact CR manifold which admits a pseudo-Einstein contact form θ with nonnegative scalar curvature. Then P ′ ≥ and thekernel of P ′ consists of the constants.Proof. It follows from (4.7) that the conclusion P ′ ≥ P ′ = R is CRinvariant, so we may compute in the scale θ . From the definition of the sublaplacianwe see that ∆ b − ∇ β ( A αβ ∇ α ) = 2 Re ∇ α (cid:0) ∇ α ∇ β ∇ β + P α (cid:1) . PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 17
It thus follows that the P ′ -operator is equivalently defined via the formula(4.10) P ′ u = 4 Re ∇ α (cid:0) ∇ α ∇ β ∇ β u − R ∇ α u (cid:1) for all u ∈ P . Multiplying (4.10) by u and integrating yields ˆ M u P ′ u = 4 ˆ M (cid:16) (cid:12)(cid:12) ∇ β ∇ β u (cid:12)(cid:12) + 2 R |∇ β u | (cid:17) . Since R ≥
0, this is clearly nonnegative, showing that P ′ ≥
0. Moreover, if equalityholds, then ∇ β ∇ β u = 0, which is easily seen to imply that u is constant, as desired. (cid:3) It would be preferable for Proposition 4.9 to require checking only CR invariantassumptions. For instance, one might hope to prove the same result assumingthat ( M , J ) has nonnegative CR Yamabe constant and admits a pseudo-Einsteincontact form. However, it is at present unclear whether these assumptions implythat one can choose a contact form as in the statement of Proposition 4.9.5. CR covariance of the P ′ -operator In this section we give a direct computational proof of transformation formula (4.7)of the P ′ -operator after a conformal change of contact form. Indeed, we will com-pute the transformation formula for the operator P ′ as defined by (4.3) acting onfunctions — rather than only pluriharmonic functions — and thus establish thatone cannot hope to find an invariant operator acting instead on the kernel of theCR Paneitz operator.To begin, we recall from Lemma 2.4 and (3.4) that given a three-dimensionalpseudohermitian manifold ( M , J, θ ) and an arbitrary one-form τ α ∈ E α ( − , − θ = e σ θ then(5.1) c ∇ α c τ α = ∇ α τ α and d W α = W α − P α σ. Another useful computation in preparation for our identification of the transfor-mation law of P ′ is the following expression for the CR Paneitz operator appliedto a product of two functions. Lemma 5.1.
Let ( M , J, θ ) be a pseudohermitian three-manifold and let P be theCR Paneitz operator. Given f, σ ∈ E (0 , , it holds that P ( f σ ) = 14 f P ( σ ) + 14 σP ( f ) + 4 Re (cid:0) ∇ α σ ∇ α ∇ β ∇ β f (cid:1) + 4 Re (cid:0) ∇ α f ∇ α ∇ β ∇ β σ (cid:1) + 2 Re ( R ∇ α σ ∇ α f ) + 2 Re (cid:0) ∇ α ∇ β σ ∇ α ∇ β f (cid:1) + 4 Re (cid:0) ∇ α ∇ α σ ∇ β ∇ β f (cid:1) . Alternatively, P ( σf ) = σP ( f ) + f P ( σ ) + 4 ∇ α σP α f + 4 ∇ α f P α σ + 4 ∇ α (cid:0) ∇ α σ ∇ β ∇ β f + ∇ β σ ∇ α ∇ β f + 2 ∇ α f ∇ β ∇ β σ + ∇ β f ∇ α ∇ β σ (cid:1) . Proof.
The second expression follows by a direct expansion using the second formulafor P in (2.5). To establish the first expression, observe that the term involving ∇ α σ in the second expression of the lemma is(5.2) ∇ α σ (cid:0) ∇ α ∇ β ∇ β f + iA αβ ∇ β f + ∇ β ∇ β ∇ α f (cid:1) . Using the assumption that M is three-dimensional and the commutator formula ∇ ¯ γ ∇ β ∇ α f − ∇ β ∇ ¯ γ ∇ α f = i ∇ ∇ α f h β ¯ γ + R αρβ ¯ γ ∇ ρ f due to Lee [25, Lemma 2.3], it holds that(5.3) ∇ β ∇ β ∇ α f − ∇ α ∇ β ∇ β f = i ∇ ∇ α f + R ∇ α f. It then follows from Lemma 2.1 that (5.2) can be rewritten ∇ α σ (cid:0) ∇ α ∇ β ∇ β f + R ∇ α f (cid:1) , from which the desired expression immediately follows. (cid:3) The transformation formulae (5.1) immediately yield the transformation formu-lae for the zeroth and first order terms of P ′ . Thus it remains to compute thetransformation formulae for the higher order terms. Proposition 5.2.
Let ( M , J, θ ) be a pseudohermitian three-manifold and definethe operator D : E (0 , → E ( − , − by Df = 4∆ b f − (cid:0) ∇ α ( A αβ ∇ β f ) (cid:1) − ∇ α ( R ∇ α f )) . If ˆ θ = e σ θ is another choice of contact form, then d Df = Df + 8 Re ∇ α (cid:0) ∇ β ∇ β σ ∇ α f + ∇ β ∇ β σ ∇ α f + ∇ α ∇ β f ∇ β σ − ∇ β ∇ β f ∇ α σ (cid:1) . Proof.
To begin, consider how each summand of D transforms. By a straightfor-ward application of Lemma 2.4, we compute that c ∆ b b f = ∆ b (∆ b f − h∇ f, ∇ σ i ) + 2 Re ∇ α ((∆ b f − h∇ f, ∇ σ i ) ∇ α σ ) c ∇ α (cid:16) d A αβ c ∇ β b f (cid:17) = ∇ α (cid:0) ( A αβ + i ∇ α ∇ β σ − i ∇ α σ ∇ β σ ) ∇ β f (cid:1)c ∇ α (cid:16) b R c ∇ α b f (cid:17) = ∇ α (cid:0) ( R + 2∆ b σ − |∇ γ σ | ) ∇ α f (cid:1) . The conclusion of the proposition then follows immediately from a straightforwardcomputation. (cid:3)
Together, (5.1), Lemma 5.1, and Proposition 5.2 yield another proof of the trans-formation law (4.7) of the P ′ -operator. In fact, the computations above allow us tocompute the transformation rule for P ′ under a change of contact form when thelocal formula (4.3) is extended to all of C ∞ ( M ). Proposition 5.3.
Let ( M , J, θ ) be a pseudohermitian three-manifold and let σ ∈ C ∞ ( M ) . Set ˆ θ = e σ θ and denote by c P ′ and P ′ the operator (4.3) defined in termsof ˆ θ and θ , respectively. Then (5.4) e σ c P ′ ( f ) = P ′ ( f ) + P ( f σ ) − σP ( f ) − P α f ∇ α σ ) for all f ∈ C ∞ ( M ) . In particular, e σ c P ′ ( f ) = P ′ ( f ) + P ( f σ ) for all f ∈ P .Remark . The transformation rule (5.4) obviously remains true when one addsa multiple of the CR Paneitz operator to P ′ . Proof.
It follows from (5.1) and Proposition 5.2 that e σ c P ′ ( f ) = P ′ ( f ) + f P σ − P α σ ∇ α f )+ 8 Re ∇ α (cid:0) ∇ β ∇ β σ )( ∇ α f ) + ( ∇ β ∇ β σ )( ∇ α f ) + ( ∇ α ∇ β f )( ∇ β σ ) − ( ∇ β ∇ β f )( ∇ α σ ) (cid:1) . PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 19
Using Lemma 5.1 to write f P σ in terms of P ( f σ ), we find that e σ c P ′ ( f ) = P ′ ( f ) + P ( σf ) − σP f − P α f ∇ α σ )+ 4 Re (cid:0) (2 ∇ α ∇ β ∇ β σ − ∇ α ∇ β ∇ β σ − ∇ β ∇ β ∇ α σ + 3 iA αβ ∇ β σ ) ∇ α f (cid:1) − (cid:0) (2 ∇ α ∇ β ∇ β f + 2 ∇ α ∇ β ∇ β f − ∇ β ∇ β ∇ α f ) ∇ α σ (cid:1) . The result then follows by using (5.3) to commute derivatives in the last two linesand the definition of the third order operator P α . (cid:3) CR transformation property of the Q ′ -curvature In this section we give a direct computational proof of the transformation for-mula (4.8) relating the Q ′ -curvatures of two pseudo-Einstein contact forms on thesame CR manifold. As in Section 5, we will in fact compute how the scalar (4.6)transforms under a conformal change of contact form without assuming either con-tact form is pseudo-Einstein. This has two benefits. First, it makes clear thatthe Q ′ -curvature only transforms as in (4.8) when both contact forms are pseudo-Einstein, as opposed to having vanishing Q -curvature. Second, it will allow usto prove Theorem 1.1 by appealing to the resolution of the CR Yamabe Prob-lem [12, 22, 23].First, as a consequence of Lemma 2.4, we see that if ˆ θ = e σ θ , then b R = R + 4 R ∆ b σ + 4(∆ b σ ) − R |∇ γ σ | − |∇ γ σ | ∆ b σ + 4 |∇ γ σ | | d A αβ | = 4 | A αβ | + 8 Im (cid:0) A αβ ∇ α ∇ β σ (cid:1) + 4 |∇ α ∇ β σ | − (cid:0) A αβ ∇ α σ ∇ β σ (cid:1) − (cid:0) ( ∇ αβ σ ) ∇ α σ ∇ β σ (cid:1) + 4 |∇ γ σ | c ∆ b b R = 2∆ b (cid:0) P + 2∆ b σ − |∇ γ σ | (cid:1) + 4 Re ∇ β (cid:0)(cid:0) R + 2∆ b σ − |∇ γ σ | (cid:1) ∇ β σ (cid:1) . It is immediately clear that the transformation law for Q ′ depends at most quadrat-ically on σ . Using the three-dimensional Bochner formula (cf. [6, 7, 10, 13]) − ∆ b |∇ γ σ | = 2 ∇ α ∇ β σ ∇ α ∇ β σ + 2 ∇ α ∇ α σ ∇ β ∇ β σ − h∇ b σ, ∇ b ∆ b σ i− (cid:0) ∇ α σ ( ∇ α ∇ β ∇ β σ − ∇ β ∇ β ∇ α σ ) (cid:1) together with the consequence12 P ( σ ) − σP ( σ ) = 8 Re ( ∇ α σP α σ ) + 8 Re (cid:0) ∇ α σ ∇ β ∇ β ∇ α σ (cid:1) + 4 ∇ α ∇ β ∇ α ∇ β σ + 8 ∇ α ∇ α σ ∇ β ∇ β σ − R |∇ γ σ | of Lemma 5.1, it follows immediately that the term of ˆ Q ′ which is quadratic in σ is given by U ( σ ) := 12 P ( σ ) − σP ( σ ) −
16 Re ( ∇ α σP α σ ) . In particular, if σ ∈ P , then U ( σ ) = 12 P ( σ ) , as expected. On the other hand, the term of ˆ Q ′ which is linear in σ is given by V ( σ ) := 4∆ b σ − (cid:0) ∇ α ( A αβ ∇ β σ ) (cid:1) − ∇ α ( R ∇ α σ )) + 8 Re ( W α ∇ α σ )= P ′ ( σ ) + 163 Re ( W α ∇ α σ ) − Qσ = P ′ ( σ ) + 163 Re ∇ α ( σW α ) + 3 Qσ.
In particular, if θ is a pseudo-Einstein contact form, then V ( σ ) = P ′ ( σ ) , as expected. In fact, we have computed the general transformation formula for thescalar invariant(6.1) Q ′ = 2∆ b R − | A αβ | + R . Proposition 6.1.
Let ( M , J, θ ) be a three-dimensional pseudohermitian manifold,regard P ′ as an operator P ′ : C ∞ ( M ) → C ∞ ( M ) , and define Q ′ by (6.1) . Givenany σ ∈ C ∞ ( M ) , the scalars Q ′ and ˆ Q ′ defined in terms of the contact forms θ and ˆ θ = e σ θ , respectively, are related by e σ ˆ Q ′ = Q ′ + P ′ ( σ ) + 163 Re ∇ α ( σW α ) + 3 Qσ + 12 P ( σ ) − σP ( σ ) −
16 Re (( ∇ α σ )( P α σ )) . (6.2) In particular, if M is compact, then (6.3) ˆ M ˆ Q ′ ˆ θ ∧ d ˆ θ = ˆ M Q ′ θ ∧ dθ + 3 ˆ M ( σP σ + 2 Qσ ) θ ∧ dθ for Q = P ′ (1) Hirachi’s Q -curvature (3.5) .Proof. (6.2) follows from the computations given above. (6.3) follows by integrationby parts. (cid:3) Proof of Theorem 1.1.
Let ˆ θ be a CR Yamabe contact form; that is, suppose thatVol ˆ θ ( M ) = 1 and R ˆ θ = Λ[ θ ] for Λ[ θ ] the CR Yamabe constant of ( M , J, θ ). Then(6.4) ˆ M ˆ R ˆ θ ∧ d ˆ θ = Λ[ θ ] ≤ Λ[ S ] for Λ[ S ] = Vol( S ) the CR Yamabe constant of the standard CR three-sphere.Moreover, by the CR Positive Mass Theorem [12], equality holds in (6.4) if andonly if ( M , J, θ ) is CR equivalent to the standard CR three-sphere. On the otherhand, the nonnegativity of the CR Paneitz operator combined with (6.3) yield(6.5) ˆ M Q ′ θ ∧ dθ ≤ ˆ M ˆ Q ′ ˆ θ ∧ d ˆ θ, while the expression (6.1) yields(6.6) ˆ M ˆ Q ′ ˆ θ ∧ d ˆ θ ≤ ˆ M ˆ R ˆ θ ∧ d ˆ θ. The result then follows from (6.4), (6.5), and (6.6). (cid:3)
PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 21
Appendix A. CR tractor bundles and the CR Paneitz operator
In this appendix we give the derivation of the CR Paneitz operator in generaldimension using tractor bundles in CR geometry, as outlined by Gover and Gra-ham [16]. In the interests of brevity, we compute in a fixed scale θ and only statethe necessary tractor formulae, and refer the reader to [16] for definitions of thetractor bundles and operators we use here.The main objects we are concerned with are the CR tractor bundle E A ∼ = E (1 , ⊕ E α (1 , ⊕ E (0 , − D opera-tor D : E ∗ ( w, w ′ ) → E A ⊗ E ∗ ( w − , w ′ ), which are given by ∇ β στ α ρ = ∇ β σ − τ β ∇ β τ α + iσA αβ ∇ β ρ − P βα τ α + σT β ∇ ¯ β στ α ρ = ∇ ¯ β σ ∇ ¯ β τ α + σP α ¯ β + ρh α ¯ β ∇ ¯ β ρ + iA α ¯ β τ α − σT ¯ β ∇ στ α ρ = ∇ σ + in +2 P σ − iρ ∇ τ α − iP αβ τ β + in +2 P τ α + 2 iσT α ∇ ρ + in +2 P ρ + 2 iT α τ α + iSσ D A f = w ( n + w + w ′ ) f ( n + w + w ′ ) ∇ α f − (cid:16) ∇ β ∇ β f + iw ∇ f + w (1 + w ′ − wn +2 ) P f (cid:17) , where E ∗ ( w, w ′ ) denotes any (weighted) tractor bundle. As always, the topmostnonvanishing slot is CR invariant. In particular, we see that the bottom row of D A D B f is the topmost nonvanishing row if n + w + w ′ = 1; as is straightforward tocheck, if we assume that f ∈ E ( w, w ′ ) for n + w + w ′ = 1, then the “bottom left”spot is the only nonvanishing term, and hence is CR invariant. More precisely, theoperator P defined by(A.1) − (cid:18) ∇ β ∇ β + i ( w − ∇ + ( w − w ′ − w + 1 n + 2 ) P (cid:19) D A f = P f will necessarily be a CR covariant operator which, as we shall see, has leading orderterm ∆ b + T (this is the reason for the factor of 4). To get the usual CR Paneitzoperator, we need to assume further that w = w ′ ; in particular, w = − n − .In order to evaluate (A.1) to determine P , we need to know the bottom compo-nents of both ∇ D A f and ∇ β ∇ β D A f . The latter is the most involved computation: Marking irrelevant terms by asterisks, we see that ∇ β ∇ β στ α ρ = h β ¯ γ ∇ ¯ γ ∇ β στ α ρ = h β ¯ γ ∇ ¯ γ ∇ β σ − τ β ∇ β τ α + iσA αβ ∇ β ρ − P αβ τ α + σT β = ∗∗∇ β ∇ β ρ − ∇ β ( P αβ τ α − σT β ) + iA αβ ∇ β τ α − σA αβ A αβ − ( ∇ β σ − τ β ) T β In particular, we see that the bottom component of ∇ β ∇ β D A f is given by − ∇ β ∇ β ( ∇ α ∇ α f + iw ∇ f + wP f ) − ∇ β (cid:0) P αβ ∇ α f − wf T β (cid:1) + iA αβ ∇ β ∇ α f − wf A αβ A αβ − ( w − T β ∇ β f. (A.2)The other derivative we must compute is ∇ D A f ; it is straightforward to checkthat the bottom component is given by − ∇ (cid:0) ∇ β ∇ β f + iw ∇ f + wP f (cid:1) − in + 2 P (cid:0) ∇ β ∇ β f + iw ∇ f + wP f (cid:1) + 2 iT α ∇ α f + iwSf. (A.3)Evaluating (A.1) using (A.2) and (A.3), we thus find that (after identifying tractorterms with their bottom components)14 P ′ f = −∇ β ∇ β D A f − i ( w − ∇ D A f − ( w − n + 3) n + 2 P D A f = ∇ β ∇ β ( ∇ α ∇ α f + iw ∇ f + wP f ) + ∇ β (cid:0) P αβ ∇ α f − wf T β (cid:1) − iA αβ ∇ β ∇ α f + wf A αβ A αβ + ( w − T β ∇ β f + i ( w − ∇ (cid:0) ∇ β ∇ β f + iw ∇ f + wP f (cid:1) + ( w − P (cid:0) ∇ β ∇ β f + iw ∇ f + wP f (cid:1) + 2( w − T α ∇ α f + w ( w − Sf.
Our goal is now to simplify this so that we can identify the CR Paneitz operator.Towards that end, let us regroup terms into those with a w coefficient and thosewithout; in other words, write(A.4) P ′ f = Af + wBf for 14 Af = ∇ β ∇ β ∇ α ∇ α f − i ∇ ∇ β ∇ β f + ∇ β (cid:0) P αβ ∇ α f (cid:1) − iA αβ ∇ β ∇ α f − T β ∇ β f − P ∇ β ∇ β f Bf = − ( w − ∇ ∇ f + i ∇ ∇ β ∇ β f + i ∇ β ∇ β ∇ f + ∇ β ∇ β ( P f ) − ∇ β ( T β f ) + A αβ A αβ f + 3 T β ∇ β f + i ( w − ∇ ( P f )+ P ∇ β ∇ β f + i ( w − P ∇ f + ( w − P f + ( w − Sf.
PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS 23
First, let us rewrite Af in a more familiar way. Using (2.2) and (2.5), it isstraightforward to check that14 Af = 14 Cf + i ( n − ∇ ∇ β ∇ β f + i ( n − A αβ ∇ β ∇ α f + in (cid:0) ∇ β A αβ (cid:1) ∇ α f + ( ∇ β P + ( n − T β ) ∇ β f + ( P α ¯ β − P h α ¯ β ) ∇ ¯ β ∇ α f − T β ∇ β f = 14 Cf + ◦ P α ¯ β ∇ ¯ β ∇ α f + n − (cid:20) i ∇ ∇ β ∇ β f + 2 i ∇ β ( A αβ ∇ α f ) − n P ∇ β ∇ β f + 4 T β ∇ β f (cid:21) . Since ◦ P α ¯ β = 0 and w = 0 when n = 1, we check in particular that P f = Cf inthis dimension.Second, recalling that w = − n − , we see from the above that Af = Cf + 4 ◦ P α ¯ β ∇ ¯ β ∇ α f + wEf Ef := − i ∇ ∇ β ∇ β f − i ∇ β ( A αβ ∇ α f ) + 2 n P ∇ β ∇ β f − T β ∇ β f. In particular, the operator F defined by F = B + E is such that P f = Cf +4 ◦ P α ¯ β ∇ ¯ β ∇ α f + wF f , and is given by14 F f = (1 − w ) ∇ ∇ f + i ∇ β ∇ β ∇ f − i ∇ ∇ β ∇ β f − i ∇ β ( A αβ ∇ α f )+ 2( n + 1) n P ∇ β ∇ β f + 2 i ( w − P ∇ f + ∇ β P ∇ β f + ∇ β P ∇ β f − T β ∇ β f − T β ∇ β f + (cid:0) ∇ β ∇ β P − ∇ β T β + i ( w − ∇ P + A αβ A αβ + ( w − P + ( w − S (cid:1) f = (1 − w ) ∇ ∇ f + i ∇ β ( A αβ ∇ α f ) − i ∇ β ( A αβ ∇ α f )+ 2( n + 1) n P ∇ β ∇ β f + 2 i ( w − P ∇ f + ∇ β P ∇ β f + ∇ β P ∇ β f − T β ∇ β f − T β ∇ β f + (cid:0) ∇ β ∇ β P − ∇ β T β + i ( w − ∇ P + A αβ A αβ + ( w − P + ( w − S (cid:1) f. Writing this entirely in terms of n , P α ¯ β , P , and A αβ then yields the desiredform. Appendix B. Checking Via the Fefferman Metric
In this appendix, we follow the other perspective of Gover and Graham [16] andgive the formula for the CR Paneitz operator using the Fefferman metric. To arriveat the formula given in Definition 4.1, we use Lee’s intrinsic formulation [24] of theFefferman metric.To begin, let ( M n +1 , J, θ ) be a pseudohermitian manifold and let ( ˜ M n +2 , g ) bethe Fefferman bundle, which is an S -bundle over M with g a particular Lorentzianmetric. The Paneitz operator L on a pseudo-Riemannian manifold is defined by L u = ∆ u + 4 P ij ∇ i ∇ j u − ( N − P ii ∆ u − ( N − ∇ j P ii )( ∇ j u ) + N − Qu, where N = 2 n + 2 is the dimension of ˜ M , P ij = N − (cid:16) R ij − N − R kk g ij (cid:17) is theSchouten tensor of g , ∆ = ∇ i ∇ i is the Laplacian (with nonpositive spectrum), and Q = − ∆ P ii − P ij P ij + N (cid:0) P ii (cid:1) is the (conformal) Q -curvature. The key facts about the Paneitz operator on theFefferman bundle are that it is conformally invariant and that its restriction tofunctions which are invariant under the circle action is itself invariant under thecircle action. In particular, these facts together imply that L descends to a CRcovariant operator on ( M n +1 , J, θ ). Explicitly, the operator P defined by(B.1) P u = 14 π ∗ ( L ( π ∗ u ))will necessarily be a CR covariant operator of the form ∆ b + T plus lower orderterms. Its explicit form can be computed using the following sequence of lemmaswhich are a consequence of Lee’s intrinsic characterization [24] of the Feffermanbundle. First, we have the following simple expressions for the scalar curvature,the Laplacian, and the inner product of two gradients on both manifolds. Lemma B.1.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold and let ( ˜ M n +2 , g ) denote the associated Fefferman bundle. Then, given any u, v ∈ C ∞ ( M ) , it holdsthat π ∗ (∆( π ∗ u )) = − b u, π ∗ J = 2 P, π ∗ h∇ ( π ∗ u ) , ∇ ( π ∗ v ) i = 4 Re ( ∇ α u ∇ α v ) . Next, we have the relationship between the norms of the Schouten tensor onboth manifolds.
Lemma B.2.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold and let ( ˜ M n +2 , g ) denote the associated Fefferman bundle. It holds that π ∗ (cid:0) P ij P ij (cid:1) = 2( n + 1) n P α ¯ β P α ¯ β + 2( n − n A αβ A αβ + 4 n ( n + 2) Im (cid:0) ∇ α ∇ β A αβ (cid:1) + 4 n ( n + 2) Re ( ∇ α ∇ α P ) . The last ingredient from [24] is the inner product of the Schouten tensor with aHessian, which follows from the formulae for the Ricci tensor and the connectionon both manifolds.
Lemma B.3.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold and let ( ˜ M n +2 , g ) denote the associated Fefferman bundle. Then, given any u ∈ C ∞ ( M ) , it holds that π ∗ (cid:0) P ij ∇ i ∇ j u (cid:1) = 4 ∇ u −
16 Im (cid:0) A αβ ∇ α ∇ β u (cid:1) +16 Re (cid:16) P α ¯ β ∇ βα u (cid:17) −
48 Re ( T α ∇ α u ) . Putting these together, we provide another derivation for the formula given inDefinition 4.1 for the CR Paneitz operator.
Proposition B.4.
Let ( M n +1 , J, θ ) be a pseudohermitian manifold and let ( ˜ M n +2 , g ) denote the associated Fefferman bundle. Denote by F and Q the operators F ( u ) = 4 P ij ∇ i ∇ j u − ( N − P kk ∆ u − ( N − ∇ j P ii )( ∇ j u ) Q = − ∆ P ii − P ij P ij + N P ii ) on ˜ M N . Then, given any u ∈ C ∞ ( M ) , it holds that π ∗ (cid:0) ∆ ( π ∗ u ) (cid:1) = ∆ b u π ∗ ( F ( π ∗ u )) = ∇ u − ∇ α (cid:0) A αβ ∇ β u (cid:1) + 4 ◦ P α ¯ β ∇ βα u − n − n Re ( ∇ α ( P ∇ α u )) − n − n ( n + 2) Re (cid:18) ( ∇ α P − in n + 1) ∇ β A αβ ) ∇ α u (cid:19) π ∗ Q = ( n + 1) n ( n + 2) ∆ b P − n ( n + 2) Im (cid:0) ∇ α ∇ β A αβ (cid:1) − n + 1 n | ◦ P α ¯ β | − n − n | A | + ( n − n + 1) n P . In particular, Definition 4.1 for the CR Paneitz operator agrees with the definitionvia (B.1) . References [1] W. Beckner. Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality.
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Department of Mathematics, Princeton University, Princeton, NJ 08540
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