## On Branson’s $Q$-curvature of order eight

HTML articles powered by AMS MathViewer

- by Andreas Juhl PDF
- Conform. Geom. Dyn.
**15**(2011), 20-43 Request permission

## Abstract:

We prove universal recursive formulas for Branson’s $Q$-curvature of order eight in terms of lower-order $Q$-curvatures, lower-order GJMS- operators and holographic coefficients. The results confirm a special case of a conjecture in [On conformally covariant powers of the Laplacian, arXiv:0905.3992v3].## References

- Helga Baum and Andreas Juhl,
*Conformal differential geometry*, Oberwolfach Seminars, vol. 40, Birkhäuser Verlag, Basel, 2010. $Q$-curvature and conformal holonomy. MR**2598414**, DOI 10.1007/978-3-7643-9909-2 - Thomas P. Branson,
*Sharp inequalities, the functional determinant, and the complementary series*, Trans. Amer. Math. Soc.**347**(1995), no. 10, 3671–3742. MR**1316845**, DOI 10.1090/S0002-9947-1995-1316845-2 - C. Falk and A. Juhl, Universal recursive formulae for $Q$-curvature. to appear in
*Crelle’s Journal*. arXiv:math/0804.2745v2. - C. Fefferman and C. R. Graham. The ambient metric. arXiv:0710.0919v2
- A. Rod Gover and Kengo Hirachi,
*Conformally invariant powers of the Laplacian—a complete nonexistence theorem*, J. Amer. Math. Soc.**17**(2004), no. 2, 389–405. MR**2051616**, DOI 10.1090/S0894-0347-04-00450-3 - A. Rod Gover and Lawrence J. Peterson,
*Conformally invariant powers of the Laplacian, $Q$-curvature, and tractor calculus*, Comm. Math. Phys.**235**(2003), no. 2, 339–378. MR**1969732**, DOI 10.1007/s00220-002-0790-4 - C. Robin Graham, Ralph Jenne, Lionel J. Mason, and George A. J. Sparling,
*Conformally invariant powers of the Laplacian. I. Existence*, J. London Math. Soc. (2)**46**(1992), no. 3, 557–565. MR**1190438**, DOI 10.1112/jlms/s2-46.3.557 - C. Robin Graham,
*Conformally invariant powers of the Laplacian. II. Nonexistence*, J. London Math. Soc. (2)**46**(1992), no. 3, 566–576. MR**1190439**, DOI 10.1112/jlms/s2-46.3.566 - C. Robin Graham,
*Volume and area renormalizations for conformally compact Einstein metrics*, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999), 2000, pp. 31–42. MR**1758076** - C. Robin Graham,
*Extended obstruction tensors and renormalized volume coefficients*, Adv. Math.**220**(2009), no. 6, 1956–1985. MR**2493186**, DOI 10.1016/j.aim.2008.11.015 - C. Robin Graham and Andreas Juhl,
*Holographic formula for $Q$-curvature*, Adv. Math.**216**(2007), no. 2, 841–853. MR**2351380**, DOI 10.1016/j.aim.2007.05.021 - C. Robin Graham and Maciej Zworski,
*Scattering matrix in conformal geometry*, Invent. Math.**152**(2003), no. 1, 89–118. MR**1965361**, DOI 10.1007/s00222-002-0268-1 - Andreas Juhl,
*Families of conformally covariant differential operators, $Q$-curvature and holography*, Progress in Mathematics, vol. 275, Birkhäuser Verlag, Basel, 2009. MR**2521913**, DOI 10.1007/978-3-7643-9900-9 - A. Juhl. On conformally covariant powers of the Laplacian. submitted. arXiv:0905.3992v3
- A. Juhl and C. Krattenthaler, Summation formulas for GJMS-operators and $Q$-curvatures on the Möbius sphere. submitted. arXiv:0910.4840v1

## Additional Information

**Andreas Juhl**- Affiliation: Humboldt-Universität, Institut für Mathematik, Unter den Linden, D-10099 Berlin, Germany
- Address at time of publication: Uppsala Universitet, Matematiska Institutionen, Box 480, S-75106 Uppsala, Sweden
- Email: andreasj@math.uu.se
- Received by editor(s): May 2, 2010
- Published electronically: March 1, 2011
- Additional Notes: This work was supported by SFB 647 “Space-Time-Matter” of DFG
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn.
**15**(2011), 20-43 - MSC (2010): Primary 53B20, 53B30; Secondary 53A30
- DOI: https://doi.org/10.1090/S1088-4173-2011-00221-9
- MathSciNet review: 2775346