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Formal theory of cornered asymptotically hyperbolic Einstein metrics. (English) Zbl 1423.53060

In this paper, the author studies the formal existence and expansion of asymptotically Einstein metrics on manifold with corner. Using the recent work of M. Nozaki et al. [J. High Energy Phys. 2012, No. 6, Paper No. 66, 25 p. (2012; Zbl 1397.81318)] the author generalizes and extends the theory of Einstein metrics on manifolds with corner and with finite boundary, showing that such metrics cannot have smooth compactifications for generic corners embedded in the infinite boundary. After that, the author studies the following. A formal expansion at the corner is derived for eigenfunctions of the scalar Laplacian subject to certain boundary conditions. Also, some important properties of the Einstein metrics are studied. Finally, it is proved that in the special case when the finite boundary is taken to be totally geodesic, there is an obstruction to existence beyond this order, which defines a conformal hypersurface invariant. In conclusion, the paper is interesting and the author presents nice and important results in the study of Einstein metrics.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B20 Local Riemannian geometry
58J37 Perturbations of PDEs on manifolds; asymptotics
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1397.81318

Software:

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References:

[1] 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) · Zbl 1307.53035 · doi:10.1007/s00526-013-0696-3
[2] Erdélyi, A.: Asymptotic Expansions. Dover, New York (1956) · Zbl 0070.29002
[3] Fefferman, C., Graham, C.R.: Conformal invariants. In: The Mathematical Heritage of Élie Cartan (Lyon, 1984). Astérisque, 95-116 (1985)
[4] Fefferman, C., Graham, C.R.: The Ambient Metric. Number 178 in Annals of Mathematics Studies. Princeton University Press, Princeton (2012) · Zbl 1243.53004
[5] Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186-225 (1991) · Zbl 0765.53034 · doi:10.1016/0001-8708(91)90071-E
[6] Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. Suppl. Rend. Circ. Mat. Palermo 63, 31-42 (2000) · Zbl 0984.53020
[7] Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89-118 (2003) · Zbl 1030.58022 · doi:10.1007/s00222-002-0268-1
[8] 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)
[9] Mazzeo, R.R.: Elliptic theory of differential edge operators I. Commun. Partial Differ. Equ. 16, 1615-1664 (1991) · Zbl 0745.58045 · doi:10.1080/03605309108820815
[10] McKeown, S.: Exponential map and normal form for cornered asymptotically hyperbolic metrics. arXiv:1609.09590, (2016) · Zbl 1454.53019
[11] Melrose, R.: Real blow up. Lecture. MSRI, (2008). http://www-math.mit.edu/ rbm/InSisp/InSiSp.html. Accessed 26 July 2018
[12] 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) · Zbl 0636.58034 · doi:10.1016/0022-1236(87)90097-8
[13] Naito, Y.: Remarks on singular Sturm comparison theorems. Mem. Differ. Equ. Math. Phys. 57, 109-122 (2012) · Zbl 1300.34071
[14] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07. Online companion to [19]
[15] Nozaki, M., Takayanagi, T., Ugajin, T.: Central charges for BCFTs and holography. J. High Energy Phys. 2012(6), 1-25 (2012) · Zbl 1397.81318 · doi:10.1007/JHEP06(2012)066
[16] Spivak, M.: A Comprehensive Introduction to Differential Geometry, 3rd edn. Publish or Perish, Houston (1999) · Zbl 1213.53001
[17] Taylor, M.: Partial Differential Equations I, 2nd edn. Springer, New York (2011) · Zbl 1206.35002 · doi:10.1007/978-1-4419-7055-8
[18] Walter, W.: Ordinary Differential Equations. Springer, New York (1998) · Zbl 0991.34001 · doi:10.1007/978-1-4612-0601-9
[19] 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) · Zbl 1198.00002
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