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


SIGMA 3 (2007), 122, 17 pages      arXiv:0712.2794      https://doi.org/10.3842/SIGMA.2007.122
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Some Progress in Conformal Geometry

Sun-Yung A. Chang a, Jie Qing b and Paul Yang a
a) Department of Mathematics, Princeton University, Princeton, NJ 08540, USA
b) Department of Mathematics, University of California, Santa Cruz, Santa Cruz, CA 95064, USA

Received August 30, 2007, in final form December 07, 2007; Published online December 17, 2007

Abstract
This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.

Key words: Bach flat metrics; bubble tree structure; degeneration of metrics; conformally compact; Einstein; renormalized volume.

pdf (254 kb)   ps (465 kb)   tex (17 kb)

References

  1. Anderson M., Orbifold compactness for spaces of Riemannian metrics and applications, Math. Ann. 331 (2005), 739-778, math.DG/0312111.
  2. Anderson M., Cheeger J., Diffeomorphism finiteness for manifolds with Ricci curvature and Ln/2-norm of curvature bounded, Geom. Funct. Anal. 1 (1991), 231-252.
  3. Brezis H., Coron J.M., Convergence of solutions of H-systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89 (1985), 21-56.
  4. Bray H., Neves A., Classification of prime 3-manifolds with Yamabe invariant larger than RP3, Ann. of Math. (2) 159 (2004), 407-424.
  5. Chang S.-Y.A., Qing J., Yang P., On a conformal gap and finiteness theorem for a class of four-manifolds. Geom. Funct. Anal. 17 (2007), 404-434, math.DG/0508621.
  6. Chang S.-Y.A., Qing J., Yang P., On the topology of conformally compact Einstein 4-manifolds, in Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, Contemp. Math. 350 (2004), 49-61, math.DG/0305085.
  7. Chang S.-Y.A., Gursky M., Yang P., An equation of Monge-Ampére type in conformal geometry and 4-manifolds of positive Ricci curvature, Ann. of Math. (2) 155 (2002), 709-787, math.DG/0409583.
  8. Chang S.-Y.A., Gursky M., Yang P., An apriori estimate for a fully nonlinear equation on 4-manifolds, J. D'Analyse Math. 87 (2002), 151-186.
  9. Graham C.R., Volume and area renormalizations for conformally compact Einstein metrics, in The Proceedings of the 19th Winter School "Geometry and Physics" (1999, Srnì), Rend. Circ. Mat. Palermo (2) 63 (2000), suppl., 31-42, math.DG/9909042.
  10. Graham C.R., Lee J., Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186-225.
  11. Qing J., On singularities of the heat flow for harmonic maps from surfaces into spheres, Comm. Anal. Geom. 3 (1995), 297-315.
  12. Qing J., On the rigidity for conformally compact Einstein manifolds, Int. Math. Res. Not. 2003 (2003), 1141-1153, math.DG/0305084.
  13. Struwe M., Global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z. 187 (1984), 511-517.
  14. Tian G., Viaclovsky J., Bach flat asymptotically ALE metrics, Invent. Math. 160 (2005), 357-415, math.DG/0310302.
  15. Tian G., Viaclovsky J., Moduli space of critical Riemannian metrics in dimension 4, Adv. Math. 196 (2005), 346-372, math.DG/0312318.


Previous article   Next article   Contents of Volume 3 (2007)