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Branson's Q-Curvature in Riemannian and Spin Geometry Branson’s Q-curvature in Riemannian and Spin Geometry Oussama Hijazi, Simon Raulot To cite this version: Oussama Hijazi, Simon Raulot. Branson’s Q-curvature in Riemannian and Spin Geometry. 2007. hal-00169447v2 HAL Id: hal-00169447 https://hal.archives-ouvertes.fr/hal-00169447v2 Preprint submitted on 1 Dec 2007 (v2), last revised 6 Feb 2008 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), ????, 14 pages Branson’s Q-curvature in Riemannian and Spin Ge- ometry Hijazi Oussama (1), Raulot Simon Institut Elie´ Cartan Nancy, Nancy-Universit´e, CNRS, INRIA, Boulevard des Aiguillettes B.P. 239 F-54506 Vandœuvre-l`es-Nancy Cedex, France E-mail: [email protected], [email protected] Received ???, in final form ????; Published online ???? Original article is available at http://www.emis.de/journals/SIGMA/2007/????/ To the memory of Tom Branson (1)Tom has deeply influenced my life. With him, I learned to push the limits of what would concretely mean to have a clear and deep thinking, to take a huge distance from things and events so that the essence could be touched. Abstract. On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson’s Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson’s Q-curvature. On a closed n-dimensional manifold, n ≥ 5, we compare the three basic conformally covariant operators : the Branson-Paneitz, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. Equality cases are also characterized. Key words: Branson’s Q-curvature; Eigenvalues; Yamabe operator; Branson-Paneitz op- erator; Dirac operator; σk-curvatures; Yamabe invariant; Conformal Geometry; Einstein metrics; Killing spinors. 2000 Mathematics Subject Classification: 53C20; 53C27; 58J50 1 Introduction The scalar curvature function behaves nicely under a conformal change of the metric. The Yamabe operator relates the scalar curvatures associated with two metrics in the same confor- mal class. From the conformal point of view dimension 2 is special and the scalar curvature corresponds to (twice) the Gauss curvature. The problem of conformally deforming the scalar curvature into a constant is known as the Yamabe problem, it has been intensively studied in the seventies and solved in the beginning eighties. hal-00169447, version 2 - 1 Dec 2007 On a spin compact manifold it is an important fact, through the Schr¨odinger-Lichnerowicz for- mula, that the scalar curvature is closely related to the eigenvalues of the Dirac operator and its sign influences the topology of the manifold. The conformal behavior of the scalar curvature and that of the Dirac operator allowed to es- tablish lower bounds for the square of the eigenvalues of the Dirac operator in terms of scalar conformal invariants, the Yamabe invariant in dimension at least 3 and the total scalar curvature in dimension 2 (see [12, 13, 4]). Given this inequality for surfaces, one might ask whether such integral inequalities could be true in any dimension. It is shown in [1] that this could not be true with the total scalar curvature. 2 O. Hijazi and S. Raulot However one could expect such integral lower bounds associated with other curvatures. The Branson Q-curvature is a scalar function which shares with the scalar curvature inter- esting conformal behavior. The operator which relates the Q-curvatures associated with two conformally related metrics is the Branson-Paneitz operator, a 4-th order differential operator. Another curvature function which is of special interest from the conformal aspect is the σk- curvature, which is the k-th symmetric function associated with the eigenvalues of the Schouten tensor (the tensor which measures the defect of the conformal invariance of the Riemann tensor). Taking into account the analogy between the scalar curvature and the Q-curvature, some natu- ral questions could be asked: what would be the role of the Q-curvature in Spin Geometry and is there any relation between the spectra of the three natural conformally covariant differential operators : Dirac, Yamabe and Branson-Paneitz? In this paper, a first answer is given for n-dimensional closed Riemannian manifolds. For n = 4, we establish a new lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson’s Q-curvature (see Theorem 2). For n ≥ 5, we show that, up to a constant, the square of the first eigenvalue of the Yamabe operator is at least equal to that of the Paneitz-Branson operator (see Theorem 3). In case the manifold is spin, we make use of what is called the Hijazi inequality to relate the first eigenvalue of the Dirac operator to the total Branson’s Q-curvature, if n = 4, and to the first eigenvalue of the Paneitz-Branson operator, if n ≥ 5. The key classical argument in Spin Geometry (see [12]) is to consider on a Riemannian manifold a special metric in the conformal class associated with an appropriate choice of the conformal factor, namely an eigenfunction associated with the first eigenvalue of the Yamabe operator. For completeness, we sketch the proof of relevant classical results in Conformal Spin Geometry and we end by introducing the notion of σk-curvature and give the proof, on a 4-dimensional closed spin manifold, of a relation between the eigenvalues of the Dirac operator and the total σ2-curvature established by G. Wang [23] . By the Chern-Gauss-Bonnet formula, it follows that the total σ2-curvature is precisely a multiple of the total Branson’s Q-curvature, hence another indirect proof of Corollary 1. 2 Natural geometric Operators in Conformal Riemannian Ge- ometry n 2u ∞ Consider a compact Riemannian manifold (M , g) and let [g] = {gu := e g / u ∈ C (M)} be the conformal class of the metric g. A class of differential operators of particular interest in Riemannian Geometry is that of conformally covariant operators. If A := Ag is a formally self-adjoint geometric differential operator acting on functions (or on sections of vector bundles) n R2 over (M , g) and Au := Agu then A is said to be conformally covariant of bidegree (a, b) ∈ if and only if −bu au Au( . )= e A(e .) (1) We now give some relevant examples of such operators. Branson’s Q-curvature in Riemannian and Spin Geometry 3 2.1 The Yamabe Operator In dimension n = 2, the Laplacian ∆ := δd acting on smooth functions is a conformally covariant −2u differential operator of bidegree (0, 2) since it satisfies ∆u = e ∆. It is interesting to note that we have the Gauss curvature equation: 2u ∆u + K = Kue , (2) 2 where Ku is the Gauss curvature of (M , gu). Using this formula, we can easily conclude that: Kdv = Kudvu, ZM ZM is a conformal invariant of the surface M equipped with the conformal class of g. In fact, it is a topological invariant due to the Gauss-Bonnet formula: 2πχ(M 2)= Kdv, (3) ZM where χ(M 2) is the Euler-Poincar´echaracteristic class of M. In dimension n ≥ 3, the Yamabe operator (or the conformal Laplacian): n − 1 L := 4 ∆+ R, (4) n − 2 n n−2 n+2 where R is the scalar curvature of (M , g), is conformally covariant of bidegree ( 2 , 2 ). Indeed, we have the following relation: n n− − +2 u 2 u Luf = e 2 L(e 2 f), (5) for all f ∈ C∞(M). It is important to note that this operator relates the scalar curvatures of the manifold M associated with two metrics in the same conformal class. Indeed, we have: n+2 n− Lu = Ru u 2 (6) 4 n− for gu = u 2 ∈ [g], where u is a smooth positive function. 2.2 The Branson-Paneitz Operator In dimension n ≥ 3, the Branson-Paneitz operator, defined by n − 4 P := ∆2 + δ(α R Id + β Ric)d + Q (7) n n 2 is a self-ajdoint elliptic fourth-order conformally covariant differential operator of bidegree n−4 n+4 ( 2 , 2 ), i.e., n n− − +4 u 4 u Puf = e 2 P (e 2 f), (8) 2 ∞ (n−2) +4 4 for all f ∈ C (M). Here αn = 2(n−1)(n−2) , βn = − n−2 , Ric is the Ricci tensor and Q is the Branson Q-curvature of (M n, g) defined by: n 1 Q = R2 − 2|S|2 + ∆R, (9) 8(n − 1)2 2(n − 1) where: 1 1 1 Sij = Aij = Ricij − Rgij . (10) n − 2 n − 2 2(n − 1) 4 O. Hijazi and S. Raulot is the Schouten tensor. The Q-curvature together with the Branson-Paneitz operator share a similar conformal behavior as that of the scalar curvature and the Yamabe operator. Indeed, if 4 n− n ≥ 5 and gu = u 4 we have: n − 4 n+4 P u = Q u n−4 , (11) 2 u n where Qu is the Q-curvature of (M , gu). On a 4-dimensional manifold, the Branson-Paneitz operator P and the Q-curvature are analogues of the Laplacian and the Gauss curvature on 2-dimensional manifolds.
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