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

Niels Martin Møller

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

Department of Mathematical Sciences

1 Citation (Scopus)

Abstract

Let (Mⁿ, g) be a closed smooth Riemannian spin manifold and denote by ∇ its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of ∇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 (Mⁿ, g) is the round n-sphere, gives a partial verification of Branson's conjecture on the pattern of extremals. Thus det∇² has a local (max, max, min, min) when the dimension is (4k, 4k+1, 4k+2, 4k+3), respectively.

Original language	English
Journal	Advances in Mathematics
Volume	229
ISSN	0001-8708
Publication status	Published - 30 Jan 2012

Cite this

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abstract = "Let (Mn, g) be a closed smooth Riemannian spin manifold and denote by ∇ its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of ∇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 (Mn, g) is the round n-sphere, gives a partial verification of Branson's conjecture on the pattern of extremals. Thus det∇2 has a local (max, max, min, min) when the dimension is (4k, 4k+1, 4k+2, 4k+3), respectively.",

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N2 - Let (Mn, g) be a closed smooth Riemannian spin manifold and denote by ∇ its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of ∇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 (Mn, g) is the round n-sphere, gives a partial verification of Branson's conjecture on the pattern of extremals. Thus det∇2 has a local (max, max, min, min) when the dimension is (4k, 4k+1, 4k+2, 4k+3), respectively.

AB - Let (Mn, g) be a closed smooth Riemannian spin manifold and denote by ∇ its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of ∇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 (Mn, g) is the round n-sphere, gives a partial verification of Branson's conjecture on the pattern of extremals. Thus det∇2 has a local (max, max, min, min) when the dimension is (4k, 4k+1, 4k+2, 4k+3), respectively.

M3 - Journal article

SN - 0001-8708

VL - 229

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

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

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