Abstract
These notes are based on my lectures at IMA, in which I tried to explain basic ideas of the ambient metric construction by studying the Szegö kernel of the sphere. The ambient metric was introduced in Fefferman [F] in his program of describing the boundary asymptotic expansion of the Bergman kernel of strictly pseudoconvex domain. This can be seen as an analogy of the description of the heat kernel asymptotic in terms of local Riemannian invariants. The counterpart of the Riemannian invariants for the Bergman kernel is invariants of the CR structure of the boundary. Thus the program consists of two parts:
-
(1)
Construct local invariants of CR structures;
-
(2)
Prove that (1) gives all invariants by using the invariant theory.
In the case of the Szegö kernel, (1) is replaced by the construction of local invariants of the Levi form that are invariant under scaling by CR pluriharmonic functions. We formulate the class of invariants in Sections 2 and 3. To simplify the presentation, we confine ourself to the case of the sphere in ℂn. It is the model case of the ambient metric construction and the basic tools already appears in this setting. We construct invariants (formulated as CR invariant differential operators) by using the ambient space in Section 4 and then explain, in Section 5, how to prove that we have got all.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T.N. Bailey, M.G. Eastwood, AND C.R. Graham, Invariant theory for conformal and CR geometry, Ann. of Math. 139 (1994), 491–552.
A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex. Studies in Advanced Math., CRC Press, 1991.
A. Čap, Infinitesimal Automorphisms and Deformations of Parabolic Geometries, arXiv:math.DG/0508535
S.S. Chern AND J.K. Moser, Real hyper surf aces in complex manifolds, Acta Math. 133 (1974), 219–271.
M.G. Eastwood AND C.R. Graham, Invariants of CR densities, in “Several complex variables and complex geometry”, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math. 52, 117–133, A.M.S., 1991.
M.G. Eastwood AND C.R. Graham, Invariants of conformal densities, Duke Math. J. 63 (1991), 633–671.
C. Fefferman, Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), 131–262.
C. Fefferman AND K. Hirachi, Ambient metric construction of Q-curvature in conformal and CR geometries, Math. Res. Lett. 10 (2003), 819–832.
A.R. Gover AND C.R. Graham, CR invariant powers of the sub-Laplacian, J. Reine Angew. Math. 583 (2005), 1–27.
C.R. Graham, R. Jenne, L.J. Mason, AND G.A.J. Sparling, Conformally invariant powers of the Laplacian. I, Existence, J. London Math. Soc. 46 (1992), 557–565.
K. Hirachi, Scalar pseudo-hermitian invariants and the Szegö kernel on three-dimensional CR manifolds, in “Complex Geometry,” Lect. Notes in Pure and Appl. Math. 143, 67–76, Dekker, 1992.
K. Hirachi, Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. 151 (2000), 151–191.
K. Hirachi, Invariant theory of the Bergman kernel of strictly pseudoconvex domains, Sugaku Expositions 17 (2004), 151–169.
J.M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), 157–178.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Hirachi, K. (2008). Ambient Metric Construction of CR Invariant Differential Operators. In: Eastwood, M., Miller, W. (eds) Symmetries and Overdetermined Systems of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 144. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73831-4_4
Download citation
DOI: https://doi.org/10.1007/978-0-387-73831-4_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-73830-7
Online ISBN: 978-0-387-73831-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)