Extremal metrics for spectral functions of Dirac operators in even and odd dimensions

TL;DR

This paper investigates the extremal properties of spectral functions of Dirac operators on Riemannian spin manifolds, establishing local extremality at stationary metrics and analyzing the spectrum of stability operators in various dimensions.

Contribution

It introduces new results on the spectrum of log-polyhomogeneous pseudodifferential operators and verifies Branson's conjecture on extremals for the determinant of Dirac operators in different dimensions.

Findings

Finiteness of Morse index at stationary metrics.

Local extremality of spectral functions under metric variations.

Verification of Branson's conjecture on extremals for round spheres.

Abstract

Let (M^n, g) be a closed smooth Riemannian spin manifold and denote by D its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of D^2, and prove finiteness of the Morse index at stationary metrics, and local extremality at such metrics under general, i.e. not only conformal, change of metrics. In even dimensions, which is also a new case for the conformal Laplacian, the relevant stability operator is of log-polyhomogeneous pseudodifferential type, and we prove new results of independent interest, on the spectrum for such operators. We use this to prove local extremality under variation of the Riemannian metric, which in the important example when (M^n, g) is the round n-sphere, gives a partial verification of Branson's conjecture on the pattern of extremals. Thus det(D^2) has a local (max, max, min, min) when the dimension is (4k, 4k + 1, 4k + 2, 4k + 3), respectively.