Sharp Inequalities, the Functional Determinant, and the Complementary Series

Abstract

Abstract.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 2, 4, and 6 for the functional determinants of operators which are well behaved under conformai change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Orsted; the method is sufficiently streamlined here that we are able to present the sixdimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformai Laplacian and of the square of the Dirac operator on S2 , and in the standard conformai classes on S4 and S6 . The S2 results are due to Onofri, and the S4 results to Branson, Chang, and Yang; the S6 results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant differential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of SOo(m + 1, 1), and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on S6 , we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding L2(S6) «-» L3(S6) for section spaces of trace free symmetric two-tensors.

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

“…The formal form ω ∞ can be realized by Borel's Lemma, 2 in the sense that there exists a form ω ∞ ∈ C ∞ (X, k (X)) with the same asymptotic expansion as ω ∞ at all orders and satisfying k ω ∞ = O(x ∞ ). Now for n even, we need to add log terms to continue the parametrix: by (2.12) one can modify ω F 1 to…”

“…n is even). A particularly interesting case is the critical one in even dimension, that is, L The main features of this operator are that it factorizes as L BG k = G BG k+1 d for some operator G 2) and that G BG k factorizes as G BG k = δ h 0 Q BG k for some differential operator are conformally invariant; the elements of H k (M) are named conformal harmonics, providing a type of Hodge theory for conformal structure. The operator Q BG k above generalizes Branson Q-curvature in the sense that it satisfies, as operators on closed k-forms,…”

“…In the case k = n/2 − 1, we have ω F 1 = ω 0 ∧ dx x + x 2 (−dδ/4 + E 1 /2) ∧ dx x , D n/2−2 = (−1) n/2+1 dE 1 /2 and B n/2−2 = (−1) n/2 dδE 1 −dE 1 δ/2−E 2 −(dδ) 2 …”

Conformally invariant powers of the Laplacian — A complete nonexistence theorem

“…If ξ is any positive function on Q homogeneous of degree −2, then ξg is independent of the action of R + on the fibres of Q, and so ξg descends to give a metric from the conformal class. Thus g determines and is equivalent to a canonical section of E ab [2] (called the conformal metric) that we also denote g (or g ab ). This in turn determines a canonical section g ab (or g −1 ) of E ab [−2] with the property that g ab g bc = δ c a (where δ a c is the kronecker delta, i.e., the section of E c a corresponding to the identity endomorphism of the tangent bundle).…”

“…Here one seeks to find a metric, from a given conformal class, that has constant scalar curvature. Recently it has become clear that higher order analogues of these operators, viz., conformally invariant operators on weighted functions (i.e., conformal densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2,5,15].…”