Conformally Invariant Powers of the Laplacian, I: Existence

On Fractional GJMS Operators

“…Given a Poincaré-Einstein manifold (X n+1 , M n , g + ), Graham and Zworski showed that for all but a finite number of values γ ∈ (0, n/2) one can define, via scattering theory, a formally self-adjoint pseudodifferential operator P 2γ on the boundary (M n , h) such that the symbol of P 2γ is the symbol of (−∆) γ and P 2γ is conformally covariant. Moreover, in the case γ ∈ N the operators P 2γ agree with the GJMS operators P 2k introduced by the eponymous Graham, Jenne, Mason and Sparling [20] via the ambient metric; in particular, P 2k is independent of the Poincaré-Einstein fill-in. For γ ∈ (0, 1), Chang and González [6] observed that, when written in terms of the compactification g := r 2 g + , where r is the geodesic defining function associated to h, the scattering definition of P 2γ becomes (1.4) P 2γ f = c γ lim γ→0 r m0 ∂U ∂r for m 0 and c γ as in (1.3) and U the function in (X, g) such that U | M = f and (1.5) div (r m0 ∇U ) + E(r, m 0 )U = 0.…”

Conformally invariant powers of the Laplacian — A complete nonexistence theorem

“…Many of these operators can be generalised to curved conformal manifolds; Graham, Jenne, Mason and Sparling [14] constructed natural conformally invariant In [13] Graham explains that "the basic reason for the nonexistence of an invariant curved modification of ∆ 3 in dimension 4 is the conformal invariance of the classical Bach tensor." An analogue of this reasoning still holds true for the proof of our theorem, although the proof is completely different from that of Graham. In higher even dimensions we replace the Bach tensor by its analogue, the FeffermanGraham obstruction tensor B ab , which arises in the ambient metric construction of [7]; see (2.9) in the next section.…”

Conformally invariant differential operators and bilinear functionals in six dimensions

“…As noted in [5], E[0] = C ∞ (M ) and E[−n] is the bundle of volume densities on M. In the 4-dimensional case, the Paneitz operator coincide with the critical GJMS operator. The GJMS operators P 2k , of Graham-Jenne-Mason-Sparling [5] by construction have the following properties.…”

“…The original work [5] uses the "ambient metric construction" of [4] to prove their existence. A computation verifies the following particular six dimensional case of Theorem 1.2.vi.…”

“…Next, we present our computation of B 4 and we prove Theorem 3.2 which characterizes any formally selfadjoint differential operator P with leading term ∆ 2 and with no zeroth-order term which is furtermore conformal invariant as a constant multiple of the Paenitz operator. After that, we present our computation of B 6 in the conformally flat case and characterize in Theorem 3.5 all the symmetric bilinear differential functionals that are conformally invariant and that define a Hochschild 2-cocycle on the algebra C ∞ (M ) for a six-dimensional manifold M. At the end, we present Theorem 4.1 which characterizes all the differential operators in dimension six that are of type GJMS [5] as explained in Section 4.…”