Extremal metrics for spectral functions of Dirac operators in even and odd dimensions. (English) Zbl 1235.58025
Let \((M^n, g)\) be a closed smooth Riemannian spin manifold. The author studies variations of the zeta-function and of the functional determinant of the associated Laplace operator under variations of the Riemannian metric \(g\). He proves finiteness of the Morse index at stationary metrics. In even dimensions the relevant stability operator is of log-polyhomogeneous pseudodifferential type, and the author proves some new results about the spectrum of such operators. The paper gives also a partial verification of Branson’s conjecture concerning the extremal properties of the standard metrics on \(S^n\).
Reviewer: Michael Farber (Zürich)
MSC:
| 58J52 | Determinants and determinant bundles, analytic torsion |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 53C24 | Rigidity results |
| 53C27 | Spin and Spin\({}^c\) geometry |
Keywords:
functional determinants; Dirac operators; spin-\(\frac 12\) fermions; renormalized one-loop effective action; spectral geometric rigidity; conformal geometry; log-polyhomogeneous pseudodifferential operatorsReferences:
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