Abstract
The ambient metric, introduced in [FG1], has proven to be an important object in conformal geometry. To a manifold M of dimension n with a conformai class of metrics [g] of signature (p, q) it associates a diffeomorphism class of formal expansions of metrics \( \tilde g \) of signature (p + 1, q + 1) on a space of dimension n + 2. This generalizes the realization of the conformal sphere Sn as the space of null lines for a quadratic form of signature (n + 1, 1), with associated Minkowski metric \( \tilde g \) on ℝn+2. The ambient space carries a family of dilations with respect to which \( \tilde g \) is homogeneous of degree 2. The other conditions determining \( \tilde g \) are that it be Ricci-flat and satisfy an initial condition specified by the conformal class [g].
This research was partially supported by NSF grant # DMS 0505701 and Grants-in-Aid for Scientific Research, JSPS.
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References
S. Alexakis, On conformally invariant differential operators in odd dimensions, Proc. Natl. Acad. Sci. USA 100 (2003), 4409–4410.
S. Alexakis, On conformally invariant differential operators, math.DG/0608771.
T.N. Bailey, M.G. Eastwood, AND A.R. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), 1191–1217.
T.N. Bailey, M.G. Eastwood, AND C.R. Graham, Invariant theory for conformal and CR geometry, Ann. Math. 139 (1994), 491–552.
T.N. Bailey AND A.R. Gover, Exceptional invariants in the parabolic invariant theory of conformal geometry, Proc. A.M.S. 123(1995), 2535–2543.
A. Čap AND A.R. Gover, Standard tractors and the conformal ambient metric construction, Ann. Global Anal. Geom. 24 (2003), 231–259.
M.G. Eastwood AND C.R. Graham, Invariants of conformal densities, Duke Math. J. 63 (1991), 633–671.
C. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. Math. 103 (1976), 395–416; Correction: Ann. Math. 104 (1976), 393–394.
C. Fefferman, Parabolic invariant theory in complex analysis, Adv. Math. 31 (1979), 131–262.
C. Fefferman AND C.R. Graham, Conformai invariants, in The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque, 1985, Numero Hors Serie, pp. 95–116.
C. Fefferman AND C.R. Graham, The ambient metric, in preparation.
A.R. Gover, Invariant theory and calculus for conformai geometries, Adv. Math. 163 (2001), 206–257.
C.R. Graham, Scalar boundary invariants and the Bergman kernel, Complex Analysis II, Proceedings, Univ. of Maryland 1985–86, Springer Lecture Notes 1276, 108–135.
C.R. Graham, Higher asymptotics of the complex Monge-Ampère equation, Comp. Math. 64 (1987), 133–155.
K. Hirachi, Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. Math. 151 (2000), 151–191.
S. Kichenassamy, On a conjecture of Fefferman and Graham, Adv. Math. 184 (2004), 268–288.
J. Lee, Higher asymptotics of the complex Monge-Ampère equation and geometry of CR-manifolds, MIT Ph.D. thesis, 1982.
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Graham, C.R., Hirachi, K. (2008). Inhomogeneous Ambient Metrics. 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_20
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