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The Tian–Yau–Zelditch Theorem and Toeplitz Operators

Part of: Partial differential equations on manifolds; differential operators Analytic continuation 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)
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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

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