Mathematics – Spectral Theory
Scientific paper
2007-05-25
Adv. Math. 229 (2012), no. 2, 1001--1046
Mathematics
Spectral Theory
45 pages; title and content edited to reflect subsequent related work
Scientific paper
10.1016/j.aim.2011.10.012
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.
Moller Niels Martin
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