Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T07:20:24.219Z Has data issue: false hasContentIssue false
  • English
  • Français

The Tian–Yau–Zelditch Theorem and Toeplitz Operators

Part of: Analytic continuation Partial differential equations on manifolds; differential operators Genetics and population dynamics

Published online by Cambridge University Press:  05 May 2011

Daniel Burns  and
Victor Guillemin
Daniel Burns
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA (dburns@umich.edu)
Victor Guillemin
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA (vwg@math.mit.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Zelditch's proof of the Tian–Yau–Zelditch Theorem makes use of the Boutet de Monvel–Sjöstrand results on the asymptotic properties of Szegö projectors for strictly pseudoconvex domains. However, as we will show below, the theorem is also closely related to another theorem of Boutet de Monvel's, namely his wave trace formula for Toeplitz operators. Finally, we will derive, for the pseudoconvex manifolds considered by Zelditch in his proof of the Tian–Yau–Zelditch Theorem, an analogue of another result of Boutet de Monvel's, the extendability theorem of Berndtsson for holomorphic functions on Grauert tubes.

Keywords

holomorphic extensionToeplitz operatorhomogeneous complex Monge–Ampère equation

MSC classification

Secondary: 32D15: Continuation of analytic objects 58J40: Pseudodifferential and Fourier integral operators on manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Special Issue 3: A Special Issue in Honour of Louis Boutet de Monvel and Pierre Schapira , July 2011 , pp. 449 - 461
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Berndtsson, B., Bergman kernels related to Hermitian line bundles over compact complex manifolds, in Explorations in complex and Riemannian geometry, pp. 1–17, Contemporary Mathematics, Volume 332 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
2.de Monvel, L. Boutet, Convergence dans le domaine complexe des séries de fonctions propres, C. R. Acad. Sci. Paris Sér. A–B 287 (1978), 855–856.Google Scholar
3.de Monvel, L. Boutet and Guillemin, V. W., Spectral properties of Toeplitz operators, Annals of Mathematics Studies, Volume 99 (Princeton University Press, 1979).Google Scholar
4.de Monvel, L. Boutet and Sjöstrand, J., Sur la singularité des noyaux de Bergman et de Szegö, in Journées: Équations aux Derivées Partielles de Rennes (1975), pp. 123–164, Astérisque, Volumes 34–35 (Société Mathématique de France, Paris, 1976).Google Scholar
5.Burns, D. and Zhang, Z., Examples of complexifications of S 2, in preparation.Google Scholar
6.Chen, X. X. and Sun, S., Space of Kähler metrics, V, Kähler quantization, preprint (arXiv:0902.4149).Google Scholar
7.de Verdière, Y. Colin, Sur les opérateurs elliptiques à bicaractéristiques toutes périodiques, Commun. Math. Helv. 54 (1979), 508–522.CrossRefGoogle Scholar
8.Donaldson, S. K., Symmetric spaces, Kähler geometry, and Hamiltonian dynamics, Am. Math. Soc. Transl. 2 196 (1999), 13–33.Google Scholar
9.Guillemin, V. W., The homogeneous Monge–Ampère equation on a pseudoconvex domain, in Méthodes Asymptotiques (Nantes, 1991), Volume 2, pp. 97–113, Astérisque, Volume 210 (Société Mathématique de France, Paris, 1992).Google Scholar
10.Guillemin, V. W., Residue traces for certain algebras of Fourier integral operators, J. Funct. Analysis 115 (1993), 391–417.CrossRefGoogle Scholar
11.Guillemin, V. W. and Stenzel, M., Grauert tubes and the homogeneous Monge–Ampère equation, J. Diff. Geom. 34 (1991), 561–570.Google Scholar
12.Guillemin, V. W. and Stenzel, M., Grauert tubes and the homogeneous Monge–Ampère equation, II, J. Diff. Geom. 35 (1992), 627–641.Google Scholar
13.Lassalle, M., Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact, Annales Scient. Éc. Norm. Sup. 11 (1978), 167–210.CrossRefGoogle Scholar
14.Lempert, L. and Szöke, R., Global solutions of the homogeneous Monge–Ampère equation and complex structures on the tangent bundle of Riemannian manifolds, Math. Annalen 290 (1991), 689–712.CrossRefGoogle Scholar
15.Mabuchi, T., Some symplectic geometry on compact Kähler manifolds, I, Osaka J. Math. 24 (1987), 227–252.Google Scholar
16.Phong, D. and Sturm, J., The Monge–Ampère operator and geodesics in the space of Kähler potentials, Invent. Math. 166 (2006), 125–149.CrossRefGoogle Scholar
17.Phong, D. and Sturm, J., Lectures on stability and constant scalar curvature, in Current developments in mathematics, 2007, pp. 101–176 (International Press, Somerville, MA, 2009).Google Scholar
18.Ruan, W.-D., Canonical coordinates and Bergman metrics, Commun. Analysis Geom. 6 (1998), 589–631.CrossRefGoogle Scholar
19.Rubinstein, Y. and Zelditch, S., The Cauchy problem for the homogeneous Monge–Ampère equation, I, Toeplitz quantization, preprint (arXiv:1008.3577 [math.DG]).Google Scholar
20.Semmes, S., Complex Monge–Ampère and symplectic manifolds, Am. J. Math. 114 (1990), 495–550.CrossRefGoogle Scholar
21.Song, J. and Zelditch, S., Convergence of Bergman geodesics on ℂℙ1, Annales Inst. Fourier 57 (2007), 2209–2237 (Special Issue: Festival Yves Colin de Verdière).CrossRefGoogle Scholar
22.Szöke, R., Complex structures on tangent bundles of Riemannian manifolds, Math. Annalen 291 (1991), 409–428.CrossRefGoogle Scholar
23.Wodzicki, M., Non-commutative residue, I, Fundamentals, in K-theory, Arithmetic and Geometry (Moscow, 1984–1986), pp. 320–399, Lecture Notes in Mathematics, Volume 1289 (Springer, 1987).Google Scholar
24.Zelditch, S., Szegö kernels and a theorem of Tian, Int. Math. Res. Not. 1998(6) (1998), 317–331.CrossRefGoogle Scholar