Results 21  30
of
54
Nonlinear Phenomena in the Spectral Theory of Geometric Linear Differential Operators
"... . The extremal problem for the functional determinant of a natural linear elliptic operator a on Riemannian manifold is studied. Viewing the determinant as a function of the Riemannian metric, we encounter nonlinear geometric analytic phenomena: sharp inequalities comparing nonlinear functionals of ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
(Show Context)
. The extremal problem for the functional determinant of a natural linear elliptic operator a on Riemannian manifold is studied. Viewing the determinant as a function of the Riemannian metric, we encounter nonlinear geometric analytic phenomena: sharp inequalities comparing nonlinear functionals of the metric and its derivatives. The derivation and use of such inequalities in new situations, especially essentially tensorvalued inequalities, leads back to linear theory and the classification of conformally covariant differential operators. 0. Introduction A central object of study in Geometric Analysis is the space G(M) of Riemannian metrics on a smooth compact manifold M . The most revealing data in this study are the spectra of differential operators A g which are functorially, or naturally, associated to the metric g; for example, the Laplacian. The diffeomorphism group Diffeo(M) is the gauge transformation group in this setting; the spectrum of a natural A g will be unaffected by ...
An Anomaly Associated With 4Dimensional Quantum Gravity
, 1996
"... . We compute the functional determinant quotient (det P h )=(det Pg ) for the Paneitz operator P in conformally related Riemannian metrics g; h, and discuss related positivity questions. 1. Introduction. In 1983, Paneitz introduced a fourthorder differential operator invariant P of conformal manifo ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
. We compute the functional determinant quotient (det P h )=(det Pg ) for the Paneitz operator P in conformally related Riemannian metrics g; h, and discuss related positivity questions. 1. Introduction. In 1983, Paneitz introduced a fourthorder differential operator invariant P of conformal manifolds, which has in many aspects proved to be the analogue of the twodimensional scalar Laplacian in fourdimensional conformal theory [P]. The original application of this operator was to conformally invariant gaugefixing for the Maxwell equations, but it has played important roles in the later studies [B13, BCY, BØ3, CY, ES12]. In [C, IV.4.fl], the functional determinant of P is found to define an anomaly associated to a quantum gravitational action defined using the Wodzicki residue [Wo]. In [BØ3], it was shown that determinant quotients for a large class of operators D which includes P are explicitly computable in dimension 4. The quotients in question have the form (det A ! )=(det A ...
Asymptotic analysis for fourth order Paneitz equations with critical growth
 Adv Calc Var
"... ar ..."
(Show Context)
The Functional Determinant
, 1993
"... Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concen ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions two, four, and six for the functional determinants of operators which are well behaved under conformal change of metric. The two dimensional formulas are due to Polyakov, and the four dimensional formulas to Branson and Ørsted; the method is sufficiently streamlined here that we are able to present the six dimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on S 2 , and in the standard conformal classes on S 4 and S 6 . The S 2 results are due to Onofri, and the S 4 result...
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 ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
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].
Pontrjagin forms and invariant objects related to the Qcurvature
 Commun. Contemp. Math
"... Abstract. We clarify the conformal invariance of the Pontrjagin forms by giving them a manifestly conformally invariant construction; they are shown to be the Pontrjagin forms of the conformally invariant tractor connection. The Qcurvature is intimately related to the Pfaffian. Working on evendime ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We clarify the conformal invariance of the Pontrjagin forms by giving them a manifestly conformally invariant construction; they are shown to be the Pontrjagin forms of the conformally invariant tractor connection. The Qcurvature is intimately related to the Pfaffian. Working on evendimensional manifolds, we show how the kform operators Qk of [10], which generalise the Qcurvature, retain a key aspect of the Qcurvature’s relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qk give rise to conformally invariant quadratic forms Θk on cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfaffian (at k = 0). Using a different construction, we show that the Qk operators yield a generalisation of the period map which maps conformal structures to Lagrangian subspaces of the direct sum H k ⊕ Hk (where Hk is the dual of the de Rham cohomology space H k). We couple the Qk operators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Qcurvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Qcurvature extends to the action functionals for these new Qlike objects. Finally we set up eigenvalue problems that generalise to Qkoperators the Qcurvature prescription problem.
Extremals for Logarithmic HLS inequalities on compact manifolds
 GAFA (To Appear
"... Abstract. Let M be a closed, compact surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g ∈ Γ, denote the area element by dV and the LaplaceBeltrami operator by ∆g. We define the Robin mass m(x) at the point x ∈ M to be the value of the G ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. Let M be a closed, compact surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g ∈ Γ, denote the area element by dV and the LaplaceBeltrami operator by ∆g. We define the Robin mass m(x) at the point x ∈ M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of ∆ −1 g is then defined by trace∆−1 = ∫ M m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let ∆V be the LaplaceBeltrami operator on the round sphere of volume V. We show that if there exists g ∈ Γ with trace ∆ −1 g < trace ∆ −1 V then the minimum of trace∆−1 over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace ∆−1 over Γ is equal to trace ∆ −1 V. In fact we prove these results in the general setting where M is an n dimensional closed, compact manifold and the LaplaceBeltrami operator is replaced by any nonnegative elliptic operator A of degree n which is conformally covariant in the sense that for the metric g we have AF2/n g = F −1Ag. In this case the role of ∆V is assumed by the Paneitz or GJMS operator on the round nsphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the OnofriBeckner theorem. Section
Constant Tcurvature conformal metric on 4manifolds with boundary
"... In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M, g) with smooth boundary there exists a metric conformal to g with constant Tcurvature, zero Qcurvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to so ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M, g) with smooth boundary there exists a metric conformal to g with constant Tcurvature, zero Qcurvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth order nonlinear elliptic boundary value problem (BVP) with boundary conditions given by a thirdorder pseudodifferential operator, and homogeneous Neumann one. It has a variational structure, but since the corresponding EulerLagrange functional is in general unbounded from below, we look for saddle points. In order to do this, we use topological arguments and minmax methods combined with a compactness result for the corresponding BVP.
On differential forms canonically associated to even dimensional conformal manifolds
"... Abstract. On a 6dimensional, conformal, oriented, compact manifold M without boundary, we compute a whole family of differential forms Ω6(f, h) of order 6, with f, h ∈ C ∞ (M). Each of these forms will be symmetric on f, and h, conformally invariant, and such that ∫ M f0Ω6(f1, f2) defines a Hochsch ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Abstract. On a 6dimensional, conformal, oriented, compact manifold M without boundary, we compute a whole family of differential forms Ω6(f, h) of order 6, with f, h ∈ C ∞ (M). Each of these forms will be symmetric on f, and h, conformally invariant, and such that ∫ M f0Ω6(f1, f2) defines a Hochschild 2cocycle over the algebra C ∞ (M). In the particular 6dimensional conformally flat case, we compute the unique one satisfying Wres(f0[F,f][F, h]) = M f0Ω6(f, h) for (H, F) the Fredholm module associated by A. Connes [6] to the manifold M, and Wres the Wodzicki residue.
EXTREMALS FOR LOGARITHMIC HARDYLITTLEWOODSOBOLEV INEQUALITIES ON COMPACT MANIFOLDS
, 2007
"... Abstract. Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g ∈ Γ, denote the area element by dV and the LaplaceBeltrami operator by ∆g. We define the Robin mass m(x) at the point x ∈ M to be the value of the ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g ∈ Γ, denote the area element by dV and the LaplaceBeltrami operator by ∆g. We define the Robin mass m(x) at the point x ∈ M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of ∆ −1 g is then defined by trace∆−1 = R M m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let ∆S2,V be the LaplaceBeltrami operator on the round sphere of volume V. We show that if there exists g ∈ Γ with trace ∆ −1 g < trace ∆ −1 S2 then the minimum of,V trace ∆−1 over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace∆−1 over Γ is equal to trace ∆ −1 S2. In fact we prove these results in the general setting where,V M is an n dimensional closed, connected manifold and the LaplaceBeltrami operator is replaced by any nonnegative elliptic operator A of degree n which is conformally covariant in the sense that for the metric g we have AF2/n g = F −1Ag. In this case the role of ∆S2,V is assumed by the Paneitz or GJMS operator on the round nsphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the OnofriBeckner theorem. Section