Sharp inequalities, the functional determinant, and the complementary series

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 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 sixdimensional 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 ${S^4}$ results to Branson, Chang, and Yang; the ${S^6}$ 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 ${\text {S}}{{\text {O}}_0}(m + 1,1)$, and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on ${S^6}$, we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding $L_1^2({S^6})\hookrightarrow {L^3}({S^6})$ for section spaces of trace free symmetric two-tensors.