Advertisement

Submanifold Conformal Invariants and a Boundary Yamabe Problem

  • A. Rod Gover
  • Andrew Waldron
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

While much is known about the invariants of conformal manifolds, the same cannot be said for the invariants of submanifolds in conformal geometries. Codimension one embedded submanifolds (or hypersurfaces) are important for applications to geometric analysis and physics.

Keywords

Conformal Invariant Conformal Structure Conformal Geometry Yamabe Problem Conformal Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Prior to this work, A.R. Gover had discussions about this problem with Fernando Marques and then Pierre Albin and Rafe Mazzeo. We are indebted for the insights so gained. The first author is supported by Marsden grant 10-UOA-113.

References

  1. 1.
    L. Andersson, P. Chrusciel, H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Commun. Math. Phys. 149, 587–612 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    P. Avilés, R.C. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds. Duke Math. J. 56, 395–398 (1988)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    P. Avilés, R.C. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds. J. Differ. Geom. 27, 225–239 (1988)MathSciNetMATHGoogle Scholar
  4. 4.
    C. Fefferman, C.R. Graham, The Ambient Metric. Annals of Mathematics Studies, vol. 178 (Princeton University Press, Princeton, 2012), x+113 ppGoogle Scholar
  5. 5.
    A.R. Gover, Almost Einstein and Poincaré–Einstein manifolds in Riemannian signature. J. Geom. Phys. 60, 182–204 (2010). arXiv:0803.3510.Google Scholar
  6. 6.
    A.R. Gover, Y. Vyatkin (in progress)Google Scholar
  7. 7.
    A.R. Gover, A. Waldron, Boundary calculus for conformally compact manifolds. Indiana Univ. Math. J. 63(1), 119–163 (2014). arXiv:1104.2991Google Scholar
  8. 8.
    C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (Academic, New York, 1974)MATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand
  2. 2.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  3. 3.Department of MathematicsUniversity of CaliforniaDavisUSA

Personalised recommendations