Abstract
This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the usual infinite boundary and a corner where the two meet. On the finite boundary, a constant mean curvature umbilic condition is imposed. First, recent work of Nozaki, Takayanagi, and Ugajin is generalized and extended showing that such metrics cannot have smooth compactifications for generic corners embedded in the infinite boundary. A model linear problem is then studied: a formal expansion at the corner is derived for eigenfunctions of the scalar Laplacian subject to certain boundary conditions. In doing so, scalar ODEs are studied that are of relevance for a broader class of boundary value problems and also for the Einstein problem. Next, unique formal existence at the corner, up to order at least equal to the boundary dimension, of Einstein metrics in a cornered asymptotically hyperbolic normal form which are polyhomogeneous in polar coordinates is demonstrated for arbitrary smooth conformal infinity. Finally it is shown that, in the special case that the finite boundary is taken to be totally geodesic, there is an obstruction to existence beyond this order, which defines a conformal hypersurface invariant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Biquard, O., Herzlich, M.: Analyse sur un demi-espace hyperbolique et poly-homogénéité locale. Calc. Var. Partial Differ. Equ. 51(3–4), 813–848 (2014)
Erdélyi, A.: Asymptotic Expansions. Dover, New York (1956)
Fefferman, C., Graham, C.R.: Conformal invariants. In: The Mathematical Heritage of Élie Cartan (Lyon, 1984). Astérisque, 95–116 (1985)
Fefferman, C., Graham, C.R.: The Ambient Metric. Number 178 in Annals of Mathematics Studies. Princeton University Press, Princeton (2012)
Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991)
Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. Suppl. Rend. Circ. Mat. Palermo 63, 31–42 (2000)
Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003)
Kontratyev, V.A.: Boundary problems for elliptic equations in domains with conical or angular points. Trudy. Mosk. Mat. Obshch 16, 209–292 (1967). (Translation, S. Feder: p. 227-314)
Mazzeo, R.R.: Elliptic theory of differential edge operators I. Commun. Partial Differ. Equ. 16, 1615–1664 (1991)
McKeown, S.: Exponential map and normal form for cornered asymptotically hyperbolic metrics. arXiv:1609.09590, (2016)
Melrose, R.: Real blow up. Lecture. MSRI, (2008). http://www-math.mit.edu/~rbm/InSisp/InSiSp.html. Accessed 26 July 2018
Mazzeo, R.R., Melrose, R.B.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75, 260–310 (1987)
Naito, Y.: Remarks on singular Sturm comparison theorems. Mem. Differ. Equ. Math. Phys. 57, 109–122 (2012)
NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07. Online companion to [19]
Nozaki, M., Takayanagi, T., Ugajin, T.: Central charges for BCFTs and holography. J. High Energy Phys. 2012(6), 1–25 (2012)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, 3rd edn. Publish or Perish, Houston (1999)
Taylor, M.: Partial Differential Equations I, 2nd edn. Springer, New York (2011)
Walter, W.: Ordinary Differential Equations. Springer, New York (1998)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)
Acknowledgements
This is doctoral work under the supervision of C. Robin Graham at the University of Washington (UW). I am most grateful to him for suggesting this and related problems, and for the really extraordinary time and attention he has given to answering questions and making suggestions large and small. I am also grateful to Andreas Karch for bringing the topic to both of our attention in the first place, to John Lee and Daniel Pollack for numerous helpful conversations, and to Hart Smith for financial support. I also greatly appreciate the excellent environment and support at Princeton University while I completed the paper. I am greatly indebted to an anonymous referee for numerous clarifications both large and small. This research was partially supported by the National Science Foundation under RTG Grants DMS-0838212 at UW and DMS-1502525 at Princeton, and Grant DMS-1161283 at UW.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by NSF RTG Grants DMS-0838212 and DMS-1502525 and Grant DMS-1161283.
Rights and permissions
About this article
Cite this article
McKeown, S.E. Formal Theory of Cornered Asymptotically Hyperbolic Einstein Metrics. J Geom Anal 29, 1876–1928 (2019). https://doi.org/10.1007/s12220-018-0067-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-018-0067-6