Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

On Branson's $ Q$-curvature of order eight


Author: Andreas Juhl
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
Published electronically: March 1, 2011
MathSciNet review: 2775346
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [BJ10] H. Baum and A. Juhl, Conformal Differential Geometry: $ Q$-curvature and Conformal Holonomy. Oberwolfach Seminars 40, 2010. MR 2598414
  • [B95] T. Branson, Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc. 347 (1995), 3671-3742. MR 1316845 (96e:58162)
  • [FJ09] C. Falk and A. Juhl, Universal recursive formulae for $ Q$-curvature. to appear in Crelle's Journal. arXiv:math/0804.2745v2.
  • [FG07] C. Fefferman and C. R. Graham. The ambient metric. arXiv:0710.0919v2
  • [GH04] A. R. Gover and K. Hirachi. Conformally invariant powers of the Laplacian--a complete nonexistence theorem. J. Amer. Math. Soc. 17, no. 2 (2004), 389-405. arXiv:math/0304082v2. MR 2051616 (2005c:58062)
  • [GP03] A. R. Gover and L. Peterson. Conformally invariant powers of the Laplacian, $ Q$-curvature, and tractor calculus. Comm. Math. Phys. 235, no. 2 (2003), 339-378. arXiv:math-ph/0201030v3. MR 1969732 (2004d:58047)
  • [GJMS92] 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. (2) 46, no. 3 (1992), 557-565. MR 1190438 (94c:58226)
  • [G92] C. R. Graham. Conformally invariant powers of the Laplacian. II. Nonexistence. J. London Math. Soc. (2) 46, no. 3 (1992), 566-576. MR 1190439 (94c:58227)
  • [G00] C. R. Graham, Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo (2) Suppl. 63 (2000), 31-42. arXiv:math/9909042v1 MR 1758076 (2002c:53073)
  • [G09] C. R. Graham, Extended obstruction tensors and renormalized volume coefficients, Adv. Math. 220, no. 6 (2009), 1956-1985. arXiv:0810.4203v1 MR 2493186 (2010e:53060)
  • [GJ07] C. R. Graham and A. Juhl, Holographic formula for $ Q$-curvature, Adv. Math., 216 (2007), 2, 841-853. arXiv:0704.1673v1 MR 2351380 (2009a:53062)
  • [GZ03] C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152, no.1 (2003), 89-118. arXiv:math/0109089v1 MR 1965361 (2004c:58064)
  • [J09a] A. Juhl. Families of Conformally Covariant Differential Operators, $ Q$-Curvature and Holography, volume 275 of Progress in Mathematics. Birkhäuser Verlag, 2009. MR 2521913 (2010m:58048)
  • [J09b] A. Juhl. On conformally covariant powers of the Laplacian. submitted. arXiv:0905.3992v3
  • [J09c] A. Juhl and C. Krattenthaler, Summation formulas for GJMS-operators and $ Q$-curvatures on the Möbius sphere. submitted. arXiv:0910.4840v1

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 53B20, 53B30, 53A30

Retrieve articles in all journals with MSC (2010): 53B20, 53B30, 53A30


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

DOI: https://doi.org/10.1090/S1088-4173-2011-00221-9
Keywords: $Q$-curvature, conformally invariant powers of Laplacian, Poincaré-Einstein metrics, renormalized volume, universality
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society