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


SIGMA 3 (2007), 120, 11 pages      arXiv:0712.2123      http://dx.doi.org/10.3842/SIGMA.2007.120
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Conformal Metrics with Constant Q-Curvature

Andrea Malchiodi
SISSA, Via Beirut 2-4, Trieste, Italy

Received September 02, 2007, in final form December 05, 2007; Published online December 13, 2007

Abstract
We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant Q-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.

Key words: Q-curvature; geometric PDEs; variational methods; min-max schemes.

pdf (261 kb)   ps (179 kb)   tex (35 kb)

References

  1. Adams D., A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), 385-398.
  2. Bahri A., Critical points at infinity in some variational problems, Research Notes in Mathematics, Vol. 182, Longman-Pitman, London, 1989.
  3. Bahri A., Coron J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294.
  4. Branson T.P., The functional determinant, Global Analysis Research Center Lecture Note Series, No 4, Seoul National University, 1993.
  5. Branson T.P., Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293-345.
  6. Branson T.P., Gover A.R., Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature, Comm. Partial Differential Equations 30 (2005), 1611-1669, math.DG/0309085.
  7. Branson T.P., Ørsted B., Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc. 113 (1991), 669-682.
  8. Branson T.P., Chang S.Y.A., Yang P.C., Estimates and extremal problems for the log-determinant on 4-manifolds, Comm. Math. Phys. 149 (1992), 241-262.
  9. Bredon G.E., Topology and geometry, Graduate Texts in Mathematics, Vol. 139, Springer, 1997.
  10. Brendle S., Global existence and convergence for a higher order flow in conformal geometry, Ann. of Math. (2) 158 (2003), 323-343, math.DG/0404415.
  11. Brendle S., Prescribing a higher order conformal invariant on Sn, Comm. Anal. Geom. 11 (2003), 837-858.
  12. Chang S.Y.A., Gursky M.J., Yang P.C., An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2) 155 (2002), 709-787, math.DG/0409583.
  13. Chang S.Y.A., Gursky M.J., Yang P.C., A conformally invariant sphere theorem in four dimensions, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 105-143, math.DG/0309287.
  14. Chang S.Y.A., Qing J., The zeta functional determinants on manifolds with boundary. I. The formula, J. Funct. Anal. 147 (1997), 327-362.
  15. Chang S.Y.A., Qing J., The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set, J. Funct. Anal. 147 (1997), 363-399.
  16. Chang S.Y.A., Qing J., Yang P.C., Compactification of a class of conformally flat 4-manifold, Invent. Math. 142 (2000), 65-93.
  17. Chang S.Y.A., Qing J., Yang P.C., On the Chern-Gauss-Bonnet integral for conformal metrics on R4, Duke Math. J. 103 (2000), 523-544.
  18. Chang S.Y.A., Yang P.C., Extremal metrics of zeta functional determinants on 4-manifolds, Ann. of Math. (2) 142 (1995), 171-212.
  19. Chang S.Y.A., Yang P.C., On a fourth order curvature invariant, in Spectral Problems in Geometry and Arithmetic, Editor T. Branson, Comtemp. Math. 237 (1999), 9-28.
  20. Chen C.C., Lin C.S., Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56 (2003), 1667-1727.
  21. Chen W., Li C., Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal. 1 (1991), 359-372.
  22. Ding W., Jost J., Li J., Wang G., Existence results for mean field equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 653-666, dg-ga/9710023.
  23. Djadli Z., Malchiodi A., A fourth order uniformization theorem on some four manifolds with large total Q-curvature, C. R. Math. Acad. Sci. Paris 340 (2005), 341-346.
  24. Djadli Z., Malchiodi A., Existence of conformal metrics with constant Q-curvature, Ann. of Math. (2), to appear, math.DG/0410141.
  25. Druet O., Robert F., Bubbling phenomena for fourth-order four-dimensional pdes with exponential growth, Proc. Amer. Math. Soc. 134 (2006), 897-908.
  26. Fefferman C., Graham C.R., Q-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), 139-151.
  27. Fefferman C., Hirachi K., Ambient metric construction of Q-curvature in conformal and CR geometries, Math. Res. Lett. 10 (2003), 819-832, math.DG/0303184.
  28. Gover A.R., Invariants and calculus for conformal geometry, Adv. Math. 163 (2001), 206-257.
  29. Gover A.R., Peterson L.J., The ambient obstruction tensor and the conformal deformation complex, Pacific J. Math. 226 (2006), 309-351, math.DG/0408229.
  30. Graham C.R., Jenne R., Mason L.J., Sparling G., Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. 46 (1992), 557-565.
  31. Graham C.R., Juhl A., Holographic formula for Q-curvature, arXiv:0704.1673.
  32. Graham C.R., Zworski M., Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89-118, math.DG/0109089.
  33. Gursky M., The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Ann. of Math. (2) 148 (1998), 315-337.
  34. Gursky M., The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Comm. Math. Phys. 207 (1999), 131-143.
  35. Gursky M., Viaclovsky J., A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom. 63 (2003) 131-154, math.DG/0301350.
  36. Li J., Li Y., Liu P., The Q-curvature on a 4-dimensional Riemannian manifold (M,g) with M QdVg = 8p2, math.DG/0608543.
  37. Malchiodi A., Compactness of solutions to some geometric fourth-order equations, J. Reine Angew. Math. 594 (2006), 137-174, math.AP/0410140.
  38. Malchiodi A., Morse theory and a scalar field equation on compact surfaces, Preprint.
  39. Malchiodi A., Struwe M., Q-curvature flow on S4, J. Differential Geom. 73 (2006), 1-44.
  40. Ndiaye C.B., Constant Q-curvature metrics in arbitrary dimension, J. Funct. Anal., to appear.
  41. Ndiaye C.B., Conformal metrics with constant Q-curvature for manifolds with boundary, Preprint, 2007.
  42. Ndiaye C.B., Constant T-curvature conformal metrics on 4-manifolds with boundary, arXiv:0708.0732.
  43. Paneitz S., A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, 1983.
  44. Struwe M., The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1988), 19-64.
  45. Struwe M., Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, 3rd ed., Springer-Verlag, Berlin, 2000.
  46. Wei J., Xu X., On conformal deformations of metrics on Sn, J. Funct. Anal 157 (1998), 292-325.

Previous article   Next article   Contents of Volume 3 (2007)