Some Remarks on the Total CR Q and Q′-Curvatures

We prove that the total CR Q-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the P′-operator and the CR invariance of the total Q′-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.

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Datum:	2018
1. Verfasser:	Marugame, T.
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Veröffentlicht:	Інститут математики НАН України 2018
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	nasplib_isofts_kiev_ua-123456789-2094542025-11-22T01:04:53Z Some Remarks on the Total CR Q and Q′-Curvatures Marugame, T. We prove that the total CR Q-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the P′-operator and the CR invariance of the total Q′-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold. The author would like to thank the referees for their comments, which were helpful for the improvement of the manuscript. 2018 Article Some Remarks on the Total CR Q and Q′-Curvatures / T. Marugame // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 32V05; 52T15 arXiv: 1711.01724 https://nasplib.isofts.kiev.ua/handle/123456789/209454 https://doi.org/10.3842/SIGMA.2018.010 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We prove that the total CR Q-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the P′-operator and the CR invariance of the total Q′-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.
format	Article
author	Marugame, T.
spellingShingle	Marugame, T. Some Remarks on the Total CR Q and Q′-Curvatures Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Marugame, T.
author_sort	Marugame, T.
title	Some Remarks on the Total CR Q and Q′-Curvatures
title_short	Some Remarks on the Total CR Q and Q′-Curvatures
title_full	Some Remarks on the Total CR Q and Q′-Curvatures
title_fullStr	Some Remarks on the Total CR Q and Q′-Curvatures
title_full_unstemmed	Some Remarks on the Total CR Q and Q′-Curvatures
title_sort	some remarks on the total cr q and q′-curvatures
publisher	Інститут математики НАН України
publishDate	2018
url	https://nasplib.isofts.kiev.ua/handle/123456789/209454
citation_txt	Some Remarks on the Total CR Q and Q′-Curvatures / T. Marugame // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 15 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT marugamet someremarksonthetotalcrqandqcurvatures
first_indexed	2025-11-28T02:53:51Z
last_indexed	2025-11-28T02:53:51Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 010, 8 pages Some Remarks on the Total CR Q and Q′-Curvatures Taiji MARUGAME Institute of Mathematics, Academia Sinica, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan E-mail: marugame@gate.sinica.edu.tw Received November 09, 2017, in final form February 12, 2018; Published online February 14, 2018 https://doi.org/10.3842/SIGMA.2018.010 Abstract. We prove that the total CR Q-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the P ′-operator and the CR invariance of the total Q′-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold. Key words: CR manifolds; Q-curvature; P ′-operator; Q′-curvature 2010 Mathematics Subject Classification: 32V05; 52T15 1 Introduction The Q-curvature, which was introduced by T. Branson [3], is a fundamental curvature quantity on even dimensional conformal manifolds. It satisfies a simple conformal transformation formula and its integral is shown to be a global conformal invariant. The ambient metric construction of the Q-curvature [9] also works for a CR manifold M of dimension 2n+ 1, and we can define the CR Q-curvature, which we denote by Q. The CR Q-curvature is a CR density of weight −n− 1 defined for a fixed contact form θ and is expressed in terms of the associated pseudo-hermitian structure. If we take another contact form θ̂ = eΥθ, Υ ∈ C∞(M), it transforms as Q̂ = Q+ PΥ, where P is a CR invariant linear differential operator, called the (critical) CR GJMS operator. Since P is formally self-adjoint and kills constant functions, the integral Q = ∫ M Q, called the total CR Q-curvature, is invariant under rescaling of the contact form and gives a global CR invariant of M . However, it follows readily from the definition of the CR Q- curvature that Q vanishes identically for an important class of contact forms, namely the pseudo- Einstein contact forms. Since the boundary of a Stein manifold admits a pseudo-Einstein contact form [5], the CR invariant Q vanishes for such a CR manifold. Moreover, it has been shown that on a Sasakian manifold the CR Q-curvature is expressed as a divergence [1], and hence Q also vanishes in this case. Thus, it is reasonable to conjecture that the total CR Q-curvature vanishes for any CR manifold, and our first result is the confirmation of this conjecture: Theorem 1.1. Let M be a compact strictly pseudoconvex CR manifold. Then the total CR Q-curvature of M vanishes: Q = 0. For three dimensional CR manifolds, Theorem 1.1 follows from the explicit formula of the CR Q-curvature; see [9]. In higher dimensions, we make use of the fact that a compact strictly pseudoconvex CR manifold M of dimension greater than three can be realized as the boundary mailto:marugame@gate.sinica.edu.tw https://doi.org/10.3842/SIGMA.2018.010 2 T. Marugame of a complex variety with at most isolated singularities [2, 10, 11]. By resolution of singularities, we can realize M as the boundary of a complex manifold X which may not be Stein. In this setting, the total CR Q-curvature is characterized as the logarithmic coefficient of the volume expansion of the asymptotically Kähler–Einstein metric on X [15]. By a simple argument using Stokes’ theorem, we prove that there is no logarithmic term in the expansion. Although the vanishing of Q is disappointing, there is an alternative Q-like object on a CR manifold which admits pseudo-Einstein contact forms. Generalizing the operator of Branson– Fontana–Morpurgo [4] on the CR sphere, Case–Yang [7] (in dimension three) and Hirachi [12] (in general dimensions) introduced the P ′-operator and the Q′-curvature for pseudo-Einstein CR manifolds. Let us denote the set of pseudo-Einstein contact forms by PE and the space of CR pluriharmonic functions by P. Two pseudo-Einstein contact forms θ, θ̂ ∈ PE are related by θ̂ = eΥθ for some Υ ∈ P. For a fixed θ ∈ PE , the P ′-operator is defined to be a linear differential operator on P which kills constant functions and satisfies the transformation formula P̂ ′f = P ′f + P (fΥ) under the rescaling θ̂ = eΥθ. The Q′-curvature is a CR density of weight −n − 1 defined for θ ∈ PE , and satisfies Q̂′ = Q′ + 2P ′Υ + P ( Υ2 ) for the rescaling. Thus, if P ′ is formally self-adjoint on P, the total Q′-curvature Q ′ = ∫ M Q′ gives a CR invariant of M . In dimension three and five, the formal self-adjointness of P ′ follows from the explicit formulas [6, 7]. In higher dimensions, Hirachi [12, Theorem 4.5] proved the formal self-adjointness under the assumption that M is the boundary of a Stein manifold X; in the proof he used Green’s formula for the asymptotically Kähler–Einstein metric g on X, and the global Kählerness of g was needed to assure that a pluriharmonic function is harmonic with respect to g. In this paper, we slightly modify his proof and prove the self-adjointness of P ′ for general pseudo-Einstein manifolds: Theorem 1.2. Let M be a compact strictly pseudoconvex CR manifold. Then the P ′-operator for a pseudo-Einstein contact form satisfies∫ M ( f1P ′f2 − f2P ′f1 ) = 0 for any f1, f2 ∈ P. Consequently, the CR invariance of Q ′ holds for any CR manifold which admits a pseudo- Einstein contact form: Theorem 1.3. Let M be a compact strictly pseudoconvex CR manifold which admits a pseudo- Einstein contact form. Then the total Q′-curvature is independent of the choice of θ ∈ PE. We note that Q ′ is a nontrivial CR invariant since it has a nontrivial variational formula; see [13]. We also give an alternative proof of Theorem 1.3 by using the characterization [12, Theorem 5.6] of Q ′ as the logarithmic coefficient in the expansion of some integral over a complex manifold with boundary M . Some Remarks on the Total CR Q and Q′-Curvatures 3 2 Proof of Theorem 1.1 We briefly review the ambient metric construction of the CR Q-curvature; we refer the reader to [9, 12, 13] for detail. Let X be an (n + 1)-dimensional complex manifold with strictly pseudoconvex CR bound- ary M , and let r ∈ C∞(X) be a boundary defining function which is positive in the interior X. The restriction of the canonical bundle KX to M is naturally isomorphic to the CR canonical bundle KM := ∧n+1(T 0,1M)⊥ ⊂ ∧n+1(CT ∗M). We define the ambient space by X̃ = KX \ {0}, and set N = KM \ {0} ∼= X̃\|M . The density bundles over X and M are defined by Ẽ(w) = ( KX ⊗KX )−w/(n+2) , E(w) = ( KM ⊗KM )−w/(n+2) ∼= Ẽ(w)\|M for each w ∈ R. We call E(w) the CR density bundle of weight w. The space of sections of Ẽ(w) and E(w) are also denoted by the same symbols. We define a C∗-action on X̃ by δλu = λn+2u for λ ∈ C∗ and u ∈ X̃. Then a section of Ẽ(w) can be identified with a function on X̃ which is homogeneous with respect to this action: Ẽ(w) ∼= { f ∈ C∞ ( X̃ ) \| δ∗λf = \|λ\|2wf for λ ∈ C∗ } . Similarly, sections of E(w) are identified with homogeneous functions on N . Let ρ ∈ Ẽ(1) be a density on X and (z1, . . . , zn+1) local holomorphic coordinates. We set ρ = \|dz1 ∧ · · · ∧ dzn+1\|2/(n+2)ρ ∈ Ẽ(0) and define J [ρ] := (−1)n+1 det ( ρ ∂zjρ ∂ziρ ∂zi∂zjρ ) . Since J [ρ] is invariant under changes of holomorphic coordinates, J defines a global differential operator, called the Monge–Ampère operator. Fefferman [8] showed that there exists ρ ∈ Ẽ(1) unique modulo O(rn+3) which satisfies J [ρ] = 1 +O(rn+2) and is a defining function of N . We fix such a ρ and define the ambient metric g̃ by the Lorentz–Kähler metric on a neighborhood of N in X̃ which has the Kähler form −i∂∂ρ. Recall that there exists a canonical weighted contact form θ ∈ Γ(T ∗M ⊗ E(1)) on M , and the choice of a contact form θ is equivalent to the choice of a positive section τ ∈ E(1), called a CR scale; they are related by the equation θ = τθ. For a CR scale τ ∈ E(1), we define the CR Q-curvature by Q = ∆̃n+1 log τ̃ \|N ∈ E(−n− 1), where ∆̃ = −∇̃I∇̃I is the Kähler Laplacian of g̃ and τ̃ ∈ Ẽ(1) is an arbitrary extension of τ . It can be shown that Q is independent of the choice of an extension of τ , and the total CR Q-curvature Q is invariant by rescaling of τ . The total CR Q-curvature has a characterization in terms of a complete metric on X. We note that the (1, 1)-form −i∂∂ logρ descends to a Kähler form on X near the boundary. We extend this Kähler metric to a hermitian metric g on X. The Kähler Laplacian ∆ = −gij∇i∇j of g is related to ∆̃ by the equation ρ∆̃f = ∆f, f ∈ Ẽ(0) (2.1) near N in X̃ \ N . In the right-hand side, we have regarded f as a function on X. For any contact form θ on M , there exists a boundary defining function ρ such that ϑ\|TM = θ, \|∂ log ρ\|g = 1 near M in X, (2.2) 4 T. Marugame where ϑ := Re(i∂ρ) ([15, Lemma 3.1]). Let ξ be the (1, 0)-vector filed on X near M characterized by ξρ = 1, ξ ⊥g H, where H := Ker ∂ρ ⊂ T 1,0X. Then, N := Re ξ is smooth up to the boundary and satisfies Nρ = 1, ϑ(N) = 0. Moreover, ν := ρN is ( √ 2)−1 times the unit outward normal vector filed along the level sets of ρ. By Green’s formula, for any function f on X we have∫ ρ>ε ∆f volg = ∫ ρ=ε νf νy volg . (2.3) Since the Monge–Ampère equation implies that g satisfies volg = −(n!)−1(1 +O(ρ))ρ−n−2dρ ∧ ϑ ∧ (dϑ)n, the formula (2.3) is rewritten as∫ ρ>ε ∆f volg = −(n!)−1 ∫ ρ=ε Nf · (1 +O(ε))ε−nϑ ∧ (dϑ)n. (2.4) With this formula, we prove the following characterization of Q. Lemma 2.1 ([15, Proposition A.3]). For an arbitrary defining function ρ, we have lp ∫ ρ>ε volg = (−1)n (n!)2(n+ 1)! Q, where lp denotes the coefficient of log ε in the asymptotic expansion in ε. Proof. Since the coefficient of log ε in the volume expansion is independent of the choice of ρ [15, Proposition 4.1], we may assume that ρ satisfies (2.2) for a fixed contact θ on M . We take τ̃ ∈ Ẽ(1) such that ρ = τ̃ ρ. Then, θ is the contact form corresponding to the CR scale τ̃ \|N . By the same argument as in the proof of [12, Lemma 3.1], we can take F ∈ Ẽ(0), G ∈ Ẽ(−n − 1) which satisfy ∆̃ ( log τ̃ + F +Gρn+1 log ρ ) = O ( ρ∞ ) , F = O(ρ), G\|N = (−1)n n!(n+ 1)! Q. We set G := τ̃n+1G ∈ Ẽ(0). By (2.1) and the equation ρ∆̃ logρ = n+ 1, we have ∆ ( log ρ− F −Gρn+1 log ρ ) = n+ 1 +O(ρ∞). Then, by using (2.4), we compute as (n+ 1) lp ∫ ρ>ε volg = lp ∫ ρ>ε ∆ ( log ρ− F −Gρn+1 log ρ ) volg = −(n!)−1 lp ∫ ρ=ε N ( log ρ− F −Gρn+1 log ρ ) · (1 +O(ε))ε−nϑ ∧ (dϑ)n = n+ 1 n! ∫ M Gθ ∧ (dθ)n = (−1)n (n!)3 Q. Thus we complete the proof. � Some Remarks on the Total CR Q and Q′-Curvatures 5 Proof of Theorem 1.1. Let ρ be an arbitrary defining function of M , and τ̃ ∈ Ẽ(1) the density on X defined by ρ = τ̃ ρ. Then α := −i∂∂ log τ̃ is a closed (1, 1)-form on X. The volume form of g is given by volg = ωn+1/(n + 1)! with the fundamental 2-form ω = igjkθ j ∧ θk. Near the boundary M in X, we have ω = −i∂∂ logρ = −i∂∂ log ρ+ α. Since the logarithmic term in the volume expansion is determined by the behavior of volg near the boundary, we compute as (n+ 1)! lp ∫ ρ>ε volg = lp ∫ ρ>ε (−i∂∂ log ρ+ α)n+1 = lp ∫ ρ>ε αn+1 + lp ∫ ρ>ε n+1∑ k=1 ( n+ 1 k ) (−i∂∂ log ρ)k ∧ αn+1−k. The first term in the last line is 0 since α is smooth up to the boundary. Using −i∂∂ log ρ = d(ϑ/ρ) and dα = 0, we also have lp ∫ ρ>ε (−i∂∂ log ρ)k ∧ αn+1−k = lp ε−k ∫ ρ=ε ϑ ∧ (dϑ)k−1 ∧ αn+1−k = 0. Thus, by Lemma 2.1 we obtain Q = 0. � 3 Proof of Theorem 1.2 We will recall the definitions of the P ′-operator and the Q′-curvature. A CR scale τ ∈ E(1) is called pseudo-Einstein if it has an extension τ̃ ∈ Ẽ(1) such that ∂∂ log τ̃ = 0 near N in X̃. The corresponding contact form θ is called a pseudo-Einstein contact form and characterized in terms of associated pseudo-hermitian structure; see [12, 13, 14]. If τ is a pseudo-Einstein CR scale, another τ̂ is pseudo-Einstein if and only if τ̂ = e−Υτ for a CR pluriharmonic function Υ ∈ P. For any f ∈ P, we take an extension f̃ ∈ Ẽ(0) such that ∂∂f̃ = 0 near M in X and define P ′f = −∆̃n+1 ( f̃ log τ̃ ) \|N ∈ E(−n− 1). We note that the germs of τ̃ and f̃ along N is unique, and P ′f is assured to be a density by ∆̃f̃ \|N = 0. The Q′-curvature is defined by Q′ = ∆̃n+1(log τ̃)2\|N ∈ E(−n− 1). Here, the homogeneity of Q′ follows from the fact ∆̃ log τ̃ \|N = 0. To prove the formal self-adjointness of P ′, we use its characterization in terms of the metric g. We define a differential operator ∆′ by ∆′f = −gij∂i∂jf . Since g is Kähler near the boundary, ∆′ agrees with ∆ near M in X. Lemma 3.1 ([12, Lemma 4.4]). Let τ ∈ E(1) be a pseudo-Einstein CR scale and τ̃ ∈ Ẽ(1) its extension such that ∂∂ log τ̃ = 0 near N in X̃. Let ρ = ρ/τ̃ be the corresponding defining function. Then, for any f ∈ C∞(X) which is pluriharmonic in a neighborhood of M in X, there exist F,G ∈ C∞(X) such that F = O(ρ) and ∆′ ( f log ρ− F −Gρn+1 log ρ ) = (n+ 1)f +O ( ρ∞ ) . Moreover, τ−n−1G\|M = (−1)n+1 (n+1)!n!P ′f holds. 6 T. Marugame In the statement of [12, Lemma 4.4], the Laplacian ∆ is used, but we may replace it by ∆′ since they agree near the boundary in X. Proof of Theorem 1.2. We extend fj to a function on X such that ∂∂fj = 0 in a neighbor- hood of M in X. Let τ be a pseudo-Einstein CR scale and ρ = ρ/τ̃ the corresponding defining function. Then we have ω = −i∂∂ log ρ near M in X. We take Fj , Gj as in Lemma 3.1 so that uj := fj log ρ−Fj−Gjρn+1 log ρ satisfies ∆′uj = (n+1)fj +O(ρ∞). We consider the coefficient of log ε in the expansion of the integral Iε = Re ∫ ρ>ε ( i∂f1 ∧ ∂u2 ∧ ωn + i∂f2 ∧ ∂u1 ∧ ωn − f1f2 ω n+1 ) , which is symmetric in the indices 1 and 2. Since dω = 0, ∂∂f2 = 0 near M in X, we have i∂f1 ∧ ∂u2 ∧ ωn = d ( if1∂u2 ∧ ωn ) − if1∂∂u2 ∧ ωn + inf1∂u2 ∧ dω ∧ ωn−1 = d ( if1∂u2 ∧ ωn ) + 1 n+ 1 f1∆′u2ω n+1 + (cpt supp), i∂f2 ∧ ∂u1 ∧ ωn = −d ( iu1∂f2 ∧ ωn ) + (cpt supp), where (cpt supp) stands for a compactly supported form on X. Thus, Iε = ∫ ρ>ε 1 n+ 1 f1 ( ∆′u2 − (n+ 1)f2 ) ωn+1 + Re ∫ ρ=ε i(f1∂u2 − u1∂f2) ∧ ωn + ∫ ρ>ε (cpt supp). The first and the third terms contain no log terms. Since ω = d(ϑ/ρ) near M in X, the second term is computed as Re ∫ ρ=ε i(f1∂u2 − u1∂f2) ∧ ωn = ε−n Re ∫ ρ=ε ( if1∂ ( f2 log ρ− F2 −G2ρ n+1 log ρ ) ∧ (dϑ)n − i ( f1 log ρ− F1 −G1ρ n+1 log ρ ) ∧ ∂f2 ∧ (dϑ)n ) +O ( ε∞ ) . The logarithmic term in the right-hand side is log ε ∫ ρ=ε (n+ 1)f1G2ϑ ∧ (dϑ)n + 2ε−n log εRe ∫ ρ=ε if1∂f2 ∧ (dϑ)n +O(ε log ε). The coefficient of log ε in the first term is (−1)n+1 (n!)2 ∫ M f1P ′f2. (3.1) The second term is equal to 2ε−n log εRe ∫ ρ>ε i∂f1 ∧ ∂f2 ∧ (dϑ)n + ε−n log ε ∫ ρ>ε (cpt supp). The first term in this formula is symmetric in the indices 1 and 2 while the second term gives no log ε term. Therefore, (3.1) should also be symmetric in 1 and 2, which implies the formal self-adjointness of P ′. � Some Remarks on the Total CR Q and Q′-Curvatures 7 4 Proof of Theorem 1.3 The formal self-adjointness of the P ′-operator implies the CR invariance of the total Q′-curva- ture. When n ≥ 2, the CR invariance can also be proved by the following characterization of Q ′ in terms of the hermitian metric g on X whose fundamental 2-form ω = igjkθ j ∧ θk agrees with −i∂∂ logρ near M in X: Theorem 4.1 ([12, Theorem 5.6]). Let τ ∈ E(1) be a pseudo-Einstein CR scale and τ̃ ∈ Ẽ(1) its extension such that ∂∂ log τ̃ = 0 near N in X̃. Let ρ = ρ/τ̃ be the corresponding defining function. Then we have lp ∫ r>ε i∂ log ρ ∧ ∂ log ρ ∧ ωn = (−1)n 2(n!)2 Q ′ (4.1) for any defining function r. In [12, Theorem 5.6], it is assumed that X is Stein and ω = −i∂∂ log ρ globally on X, but as the logarithmic term is determined by the boundary behavior, it is sufficient to assume ω = −i∂∂ log ρ near M in X as above. Proof of Theorem 1.3. Let τ , ρ be as in Theorem 4.1 and let ρ̂ be the defining function corresponding to another pseudo-Einstein CR scale τ̂ . Then we can write as ρ̂ = eΥρ with Υ ∈ C∞(X) such that ∂∂Υ = 0 near M in X. Using the defining function ρ for r in the formula (4.1), we compute as lp ∫ ρ>ε i∂ log ρ̂ ∧ ∂ log ρ̂ ∧ ωn = lp ∫ ρ>ε i(∂ log ρ+ ∂Υ) ∧ (∂ log ρ+ ∂Υ) ∧ ωn = lp ∫ ρ>ε i∂ log ρ ∧ ∂ log ρ ∧ ωn + lp ∫ ρ>ε i∂Υ ∧ ∂Υ ∧ ωn + 2 Re lp ∫ ρ>ε i∂ log ρ ∧ ∂Υ ∧ ωn. The second term in the last line is lp ∫ ρ>ε i∂Υ ∧ ∂Υ ∧ ωn = lp ∫ ρ=ε iΥ∂Υ ∧ ωn + lp ∫ ρ>ε (cpt supp) = 0. Since ω = d(ϑ/ρ) near M in X, we have∫ ρ>ε i∂ log ρ ∧ ∂Υ ∧ ωn = log ε ∫ ρ=ε i∂Υ ∧ ωn + ∫ ρ>ε (cpt supp) = ε−n log ε ∫ ρ=ε i∂Υ ∧ (dϑ)n + ∫ ρ>ε (cpt supp) = ε−n log ε ∫ ρ>ε (cpt supp) + ∫ ρ>ε (cpt supp), which implies that the third term is also 0. Thus, Q ′ is independent of the choice of a pseudo- Einstein CR scale τ . � Acknowledgements The author would like to thank the referees for their comments which were helpful for the improvement of the manuscript. 8 T. Marugame References [1] Alexakis S., Hirachi K., Integral Kähler invariants and the Bergman kernel asymptotics for line bundles, Adv. 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Appl. 25 (2007), 356–379, math.DG/0404455. https://doi.org/10.1016/j.aim.2016.12.014 https://arxiv.org/abs/1501.02463 https://doi.org/10.4007/annals.2013.177.1.1 https://arxiv.org/abs/0712.3905 https://doi.org/10.1512/iumj.2007.56.3111 https://arxiv.org/abs/math.DG/0609312 https://arxiv.org/abs/1709.08057 https://arxiv.org/abs/1309.2528 https://doi.org/10.2307/1970945 https://doi.org/10.4310/MRL.2003.v10.n6.a9 https://arxiv.org/abs/math.DG/0303184 https://doi.org/10.2307/1971032 https://doi.org/10.2307/1971093 https://doi.org/10.1016/j.difgeo.2013.10.013 https://arxiv.org/abs/1302.0489 https://doi.org/10.1016/j.aim.2016.11.005 https://doi.org/10.1016/j.aim.2016.11.005 https://arxiv.org/abs/1510.03221 https://doi.org/10.2307/2374543 https://doi.org/10.1016/j.difgeo.2007.02.004 https://arxiv.org/abs/math.DG/0404455 1 Introduction 2 Proof of Theorem 1.1 3 Proof of Theorem 1.2 4 Proof of Theorem 1.3 References

Some Remarks on the Total CR Q and Q′-Curvatures

Institution

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