Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 010, 8 pages      arXiv:1711.01724

Some Remarks on the Total CR $Q$ and $Q^\prime$-Curvatures

Taiji Marugame
Institute of Mathematics, Academia Sinica, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan

Received November 09, 2017, in final form February 12, 2018; Published online February 14, 2018

We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.

Key words: CR manifolds; $Q$-curvature; $P^\prime$-operator; $Q^\prime$-curvature.

pdf (313 kb)   tex (13 kb)


  1. Alexakis S., Hirachi K., Integral Kähler invariants and the Bergman kernel asymptotics for line bundles, Adv. Math. 308 (2017), 348-403, arXiv:1501.02463.
  2. Boutet de Monvel L., Intégration des équations de Cauchy-Riemann induites formelles, in Séminaire Goulaouic-Lions-Schwartz 1974-1975; Équations aux derivées partielles linéaires et non linéaires, Centre Math., École Polytech., Paris, 1975, Exp. No. 9, 14 pages.
  3. Branson T.P., The functional determinant, Lecture Notes Series, Vol. 4, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993.
  4. Branson T.P., Fontana L., Morpurgo C., Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Ann. of Math. 177 (2013), 1-52, arXiv:0712.3905.
  5. Cao J., Chang S.C., Pseudo-Einstein and $Q$-flat metrics with eigenvalue estimates on CR-hypersurfaces, Indiana Univ. Math. J. 56 (2007), 2839-2857, math.DG/0609312.
  6. Case J.S., Gover A.R., The $P^\prime$-operator, the $Q^\prime$-curvature, and the CR tractor calculus, arXiv:1709.08057.
  7. Case J.S., Yang P., A Paneitz-type operator for CR pluriharmonic functions, Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013), 285-322, arXiv:1309.2528.
  8. Fefferman C.L., Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. 103 (1976), 395-416.
  9. Fefferman C.L., Hirachi K., Ambient metric construction of $Q$-curvature in conformal and CR geometries, Math. Res. Lett. 10 (2003), 819-831, math.DG/0303184.
  10. Harvey F.R., Lawson Jr. H.B., On boundaries of complex analytic varieties. I, Ann. of Math. 102 (1975), 223-290.
  11. Harvey F.R., Lawson Jr. H.B., On boundaries of complex analytic varieties. II, Ann. of Math. 106 (1977), 213-238.
  12. Hirachi K., $Q$-prime curvature on CR manifolds, Differential Geom. Appl. 33 (2014), suppl., 213-245, arXiv:1302.0489.
  13. Hirachi K., Marugame T., Matsumoto Y., Variation of total $Q$-prime curvature on CR manifolds, Adv. Math. 306 (2017), 1333-1376, arXiv:1510.03221.
  14. Lee J.M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), 157-178.
  15. Seshadri N., Volume renormalization for complete Einstein-Kähler metrics, Differential Geom. Appl. 25 (2007), 356-379, math.DG/0404455.

Previous article  Next article   Contents of Volume 14 (2018)