American Institute of Mathematical Sciences

Advanced
October  2008, 2(4): 701-718. doi: 10.3934/jmd.2008.2.701

On the spectrum of geometric operators on Kähler manifolds

Dmitry Jakobson 1, , Alexander Strohmaier 2, and  Steve Zelditch 3,

1. 

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6, Canada
2. 

Department of Mathematical Sciences, LoughboroughUniversity, Loughborough, Leicestershire, LE11 3TU, United Kingdom
3. 

Johns Hopkins University, Department of Mathematics, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, United States

Received  May 2008 Revised  September 2008 Published  October 2008

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: Kähler manifold, frame flow, Dirac operator, quantum ergodicity, eigenfunction.
Mathematics Subject Classification: Primary: 81Q50 Secondary: 35P20, 37D30, 58J50, 81Q005.
Citation: Dmitry Jakobson, Alexander Strohmaier, Steve Zelditch. On the spectrum of geometric operators on Kähler manifolds. Journal of Modern Dynamics, 2008, 2 (4) : 701-718. doi: 10.3934/jmd.2008.2.701
[1]

Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137
[2]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
[3]

Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287
[4]

Carlos Kenig, Tobias Lamm, Daniel Pollack, Gigliola Staffilani, Tatiana Toro. The Cauchy problem for Schrödinger flows into Kähler manifolds. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 389-439. doi: 10.3934/dcds.2010.27.389
[5]

Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97.

[6]

Jundong Zhou. A class of the non-degenerate complex quotient equations on compact Kähler manifolds. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2361-2377. doi: 10.3934/cpaa.2021085
[7]

Dmitry Jakobson and Alexander Strohmaier. High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows. Electronic Research Announcements, 2006, 12: 87-94.

[8]

Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015
[9]

Toshihiro Iwai, Dmitrií A. Sadovskií, Boris I. Zhilinskií. Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries. Journal of Geometric Mechanics, 2020, 12 (3) : 455-505. doi: 10.3934/jgm.2020021
[10]

David Constantine. 2-Frame flow dynamics and hyperbolic rank-rigidity in nonpositive curvature. Journal of Modern Dynamics, 2008, 2 (4) : 719-740. doi: 10.3934/jmd.2008.2.719
[11]

Habibul Islam, Om Prakash, Ram Krishna Verma. New quantum codes from constacyclic codes over the ring $ R_{k,m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020097
[12]

Nigel Higson and Gennadi Kasparov. Operator K-theory for groups which act properly and isometrically on Hilbert space. Electronic Research Announcements, 1997, 3: 131-142.

[13]

Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086
[14]

Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019
[15]

Qiao-Fang Lian, Yun-Zhang Li. Reducing subspace frame multiresolution analysis and frame wavelets. Communications on Pure & Applied Analysis, 2007, 6 (3) : 741-756. doi: 10.3934/cpaa.2007.6.741
[16]

Yvette Kosmann-Schwarzbach. Dirac pairs. Journal of Geometric Mechanics, 2012, 4 (2) : 165-180. doi: 10.3934/jgm.2012.4.165
[17]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675
[18]

Yaiza Canzani, Boris Hanin. Fixed frequency eigenfunction immersions and supremum norms of random waves. Electronic Research Announcements, 2015, 22: 76-86. doi: 10.3934/era.2015.22.76
[19]

Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845
[20]

Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy