A Construction of Critical GJMS Operators Using Wodzicki's Residue

Abstract

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical GJMS operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order −2, originally defined by A. Connes, acting on middle dimension forms.

Conformally invariant differential operators and bilinear functionals in six dimensions

“…A computation verifies the following particular six dimensional case of Theorem 1.2.vi. of [12] Proposition 4.2. The functional B 6 and the polynomial P 6 are related by P 6 (fh) = P 6 (f )h + fP 6 (h) − 2B 6 (f, h).…”

“…Furthermore, by Lemma 3.2 [12], we know that B n (f, h) is a polynomial in the ingredients ∇ α df, ∇ β dh, and ∇ γ R for multi-indices α, β, and γ, with 2 deg…”

“…Stokes' theorem applied to (14) leads to the expression [12] for details). Because this equality holds for every f and every h in C ∞ (M ) we obtain an operator P 6 constructed in a canonical way from the Wodzicki residue Wres ([F, f ][F, h]).…”

“…E-Mail: wugalde@cariari.ucr.ac.cr with k, f, h ∈ C ∞ (M ). For details in the four-dimensional case see [3] and for the general case see [12]. Here Wres is the non-commutative residue.…”

Some Conformal Invariants from the Noncommutative Residue for Manifolds with Boundary

“…There is one particular aspect of noncommutative geometry that has historically received less attention than other of its subjects; the use of its machinery to obtain conformal invariants (associated to the underlying manifold). The motivating example in this venue, is a conformal invariant in dimension 4 (Connes [5]) and its extension to higher order even dimensional manifolds by the author [19]. The main idea lies on Theorem IV.4.2.c of Connes [6].…”