Journal of Modern Dynamics (JMD)

On the spectrum of geometric operators on Kähler manifolds

Pages: 701 - 718, Issue 4, October 2008      doi:10.3934/jmd.2008.2.701

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Dmitry Jakobson - Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6, Canada (email)
Alexander Strohmaier - Department of Mathematical Sciences, LoughboroughUniversity, Loughborough, Leicestershire, LE11 3TU, United Kingdom (email)
Steve Zelditch - Johns Hopkins University, Department of Mathematics, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, United States (email)

Abstract: On a compact Kähler manifold, there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator, and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace--Beltrami operator. Because of the high degree of symmetry, the Laplace--Beltrami operator on forms can not be quantum ergodic. We show that, after taking these symmetries into account, quantum ergodicity holds for the Laplace--Beltrami operator and for the Spin$^\cbb$-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved Kähler manifolds of odd complex dimension.

Keywords:  Dirac operator, eigenfunction, frame flow, quantum ergodicity, Kähler manifold
Mathematics Subject Classification:  Primary: 81Q50 Secondary: 35P20, 37D30, 58J50, 81Q005

Received: May 2008;      Revised: September 2008;      Available Online: October 2008.