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11
ON CONFORMALLY COVARIANT POWERS OF THE LAPLACIAN
, 2009
"... We propose and discuss recursive formulae for conformally covariant powers P2N of the Laplacian (GJMSoperators). For locally conformally flat metrics, these describe the nonconstant part of any GJMSoperator as the sum of a certain linear combination of compositions of lower order GJMSoperators ..."
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Cited by 6 (4 self)
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We propose and discuss recursive formulae for conformally covariant powers P2N of the Laplacian (GJMSoperators). For locally conformally flat metrics, these describe the nonconstant part of any GJMSoperator as the sum of a certain linear combination of compositions of lower order GJMSoperators (primary part) and a secondorder operator which is defined by the Schouten tensor (secondary part). We complete the description of GJMSoperators by proposing and discussing recursive formulae for their constant terms, i.e., for Branson’s Qcurvatures, along similar lines. We confirm the picture in a number of cases. Full proofs are given for spheres of any dimension and arbitrary signature. Moreover, we prove formulae of the respective critical third power P6 in terms of the Yamabe operator P2 and the Paneitz operator P4, and of a fourth power in terms of P2, P4 and P6. For general metrics, the latter involves the first two of Graham’s extended obstruction tensors [G4]. In full generality, the recursive formulae remain conjectural. We describe their relation to the theory of residue families and the associated Qpolynomials as developed in [J1].
CALCULUS AND INVARIANTS ON ALMOST COMPLEX MANIFOLDS, INCLUDING PROJECTIVE AND CONFORMAL GEOMETRY
"... Abstract. We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular, we are able to construct an invariant and effi ..."
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Abstract. We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular, we are able to construct an invariant and efficient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case, we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting, we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants. 1.
THE PROFILE OF BUBBLING SOLUTIONS OF A CLASS OF FOURTH ORDER GEOMETRIC EQUATIONS ON 4MANIFOLDS
, 2008
"... We study a class of fourth order geometric equations defined on a 4dimensional compact Riemannian manifold which includes the Qcurvature equation. We obtain sharp estimates on the difference near the blowup points between a bubbling sequence of solutions and the standard bubble. ..."
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Cited by 2 (0 self)
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We study a class of fourth order geometric equations defined on a 4dimensional compact Riemannian manifold which includes the Qcurvature equation. We obtain sharp estimates on the difference near the blowup points between a bubbling sequence of solutions and the standard bubble.
Universal recursive formulae for Qcurvature
"... Abstract. We discuss recursive formulas for Branson’s Qcurvatures. The formulas present Qcurvatures of any order in terms of lower order Qcurvatures and lower order GJMSoperators. These presentations are universal in the sense that the recursive structure does not depend on the dimension of the ..."
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Cited by 2 (2 self)
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Abstract. We discuss recursive formulas for Branson’s Qcurvatures. The formulas present Qcurvatures of any order in terms of lower order Qcurvatures and lower order GJMSoperators. These presentations are universal in the sense that the recursive structure does not depend on the dimension of the underlying space. We give proofs for Q4 and Q6 for general metrics, and for Q8 for conformally flat metrics. The general case is conjectural. We display explicit formulas for Qcurvatures of order up to 16. The high order cases are tested for round spheres of even dimension and Einstein metrics. A part of the structure of the universal recursive formulas is described in terms of a generating function. The results rest on the theory of residue families ([25]).
Existence and Concentration of Positive Solutions for a Supercritical Fourth Order Equation
"... Abstract In this paper we investigate the problem of existence of solutions for a supercritical fourth order Yamabe type equation and we exhibit a family of solutions concentrating at two points, provided the domain contains one hole and we give a multiplicity result if we are given multiple holes ..."
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Abstract In this paper we investigate the problem of existence of solutions for a supercritical fourth order Yamabe type equation and we exhibit a family of solutions concentrating at two points, provided the domain contains one hole and we give a multiplicity result if we are given multiple holes.
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"... Abstract. We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an a ne connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and e cient ..."
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Abstract. We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an a ne connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and e cient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we nd a projectively invariant tensor the vanishing of which is necessary and su cient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants.
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, 2008
"... 1.2 Goals and objectives......................... 2 1.3 Related work conducted at DFKI.................. 2 ..."
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1.2 Goals and objectives......................... 2 1.3 Related work conducted at DFKI.................. 2
CONFORMAL INVARIANTS FROM NODAL SETS. I. NEGATIVE EIGENVALUES AND CURVATURE PRESCRIPTION
"... In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a ..."
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In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant’s Nodal Domain Theorem. We also show that on any manifold of dimension n ≥ 3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n ≥ 3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to leftinvariant metrics on Heisenberg manifolds. Finally, in the appendix, the 2nd named author and Andrea Malchiodi study the Qcurvature prescription problems for noncritical Qcurvatures.