An invariant related to the existence of conformally compact Einstein fillings

Gursky, Matthew; Han, Qing; Stolz, Stephan

doi:10.1090/tran/8308

An invariant related to the existence of conformally compact Einstein fillings
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by Matthew J. Gursky, Qing Han and Stephan Stolz PDF

Trans. Amer. Math. Soc. 374 (2021), 4185-4205 Request permission

Abstract:

We define an invariant for compact spin manifolds $X$ of dimension $4k$ equipped with a metric $h$ of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of $h$ to be the conformal infinity of a conformally compact Einstein metric on $X$.

References

Michael T. Anderson, Geometric aspects of the AdS/CFT correspondence, AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lect. Math. Theor. Phys., vol. 8, Eur. Math. Soc., Zürich, 2005, pp. 1–31. MR 2160865, DOI 10.4171/013-1/1
Michael T. Anderson, Einstein metrics with prescribed conformal infinity on 4-manifolds, Geom. Funct. Anal. 18 (2008), no. 2, 305–366. MR 2421542, DOI 10.1007/s00039-008-0668-5
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
Olivier Biquard, Métriques d’Einstein asymptotiquement symétriques, Astérisque 265 (2000), vi+109 (French, with English and French summaries). MR 1760319
Olivier Biquard, Métriques autoduales sur la boule, Invent. Math. 148 (2002), no. 3, 545–607 (French, with English summary). MR 1908060, DOI 10.1007/s002220100203
Olivier Biquard, Désingularisation de métriques d’Einstein. II, Invent. Math. 204 (2016), no. 2, 473–504 (French, with English summary). MR 3489703, DOI 10.1007/s00222-015-0619-3
Bernhelm Booß-Bavnbek and Krzysztof P. Wojciechowski, Elliptic boundary problems for Dirac operators, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1233386, DOI 10.1007/978-1-4612-0337-7
David M. J. Calderbank and Michael A. Singer, Einstein metrics and complex singularities, Invent. Math. 156 (2004), no. 2, 405–443. MR 2052611, DOI 10.1007/s00222-003-0344-1
Sun-Yung A. Chang, Jie Qing, and Paul Yang, On the topology of conformally compact Einstein 4-manifolds, Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence, RI, 2004, pp. 49–61. MR 2082390, DOI 10.1090/conm/350/06337
Piotr T. Chruściel and Erwann Delay, Non-singular, vacuum, stationary space-times with a negative cosmological constant, Ann. Henri Poincaré 8 (2007), no. 2, 219–239. MR 2314449, DOI 10.1007/s00023-006-0306-4
Piotr T. Chruściel and Erwann Delay, Non-singular space-times with a negative cosmological constant: II. Static solutions of the Einstein-Maxwell equations, Lett. Math. Phys. 107 (2017), no. 8, 1391–1407. MR 3669238, DOI 10.1007/s11005-017-0955-x
Piotr T. Chruściel, Erwann Delay, John M. Lee, and Dale N. Skinner, Boundary regularity of conformally compact Einstein metrics, J. Differential Geom. 69 (2005), no. 1, 111–136. MR 2169584
Xianzhe Dai and Guofang Wei, Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds, Adv. Math. 214 (2007), no. 2, 551–570. MR 2349712, DOI 10.1016/j.aim.2007.02.010
Tohru Eguchi, Peter B. Gilkey, and Andrew J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980), no. 6, 213–393. MR 598586, DOI 10.1016/0370-1573(80)90130-1
Charles Fefferman and C. Robin Graham, Conformal invariants, Astérisque Numéro Hors Série (1985), 95–116. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837196
PawełGajer, Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. Global Anal. Geom. 5 (1987), no. 3, 179–191. MR 962295, DOI 10.1007/BF00128019
Peter B. Gilkey, The index theorem and the heat equation, Mathematics Lecture Series, No. 4, Publish or Perish, Inc., Boston, Mass., 1974. Notes by Jon Sacks. MR 0458504
C. Robin Graham and John M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), no. 2, 186–225. MR 1112625, DOI 10.1016/0001-8708(91)90071-E
Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423–434. MR 577131, DOI 10.2307/1971103
Mikhael Gromov and H. Blaine Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984). MR 720933
Matthew J. Gursky and Qing Han, Non-existence of Poincaré-Einstein manifolds with prescribed conformal infinity, Geom. Funct. Anal. 27 (2017), no. 4, 863–879. MR 3678503, DOI 10.1007/s00039-017-0414-y
S. W. Hawking and Don N. Page, Thermodynamics of black holes in anti-de Sitter space, Comm. Math. Phys. 87 (1982/83), no. 4, 577–588. MR 691045
Oussama Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), no. 1, 151–162. MR 834486
Oussama Hijazi, Sebastián Montiel, and Xiao Zhang, Eigenvalues of the Dirac operator on manifolds with boundary, Comm. Math. Phys. 221 (2001), no. 2, 255–265. MR 1845323, DOI 10.1007/s002200100475
Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 358873, DOI 10.1016/0001-8708(74)90021-8
D. D. Joyce, Compact $8$-manifolds with holonomy $\textrm {Spin}(7)$, Invent. Math. 123 (1996), no. 3, 507–552. MR 1383960, DOI 10.1007/s002220050039
John M. Lee, The spectrum of an asymptotically hyperbolic Einstein manifold, Comm. Anal. Geom. 3 (1995), no. 1-2, 253–271. MR 1362652, DOI 10.4310/CAG.1995.v3.n2.a2
John M. Lee, Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc. 183 (2006), no. 864, vi+83. MR 2252687, DOI 10.1090/memo/0864
André Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9 (French). MR 156292
Juan Maldacena, The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), no. 2, 231–252. MR 1633016, DOI 10.4310/ATMP.1998.v2.n2.a1
Fernando Codá Marques, Deforming three-manifolds with positive scalar curvature, Ann. of Math. (2) 176 (2012), no. 2, 815–863. MR 2950765, DOI 10.4007/annals.2012.176.2.3
Rafe Mazzeo and Frank Pacard, Maskit combinations of Poincaré-Einstein metrics, Adv. Math. 204 (2006), no. 2, 379–412. MR 2249618, DOI 10.1016/j.aim.2005.06.001
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Peyman Morteza and Jeff A. Viaclovsky, The Calabi metric and desingularization of Einstein orbifolds, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 4, 1201–1245. MR 4071325, DOI 10.4171/JEMS/943
Jonathan Rosenberg, Manifolds of positive scalar curvature: a progress report, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 259–294. MR 2408269, DOI 10.4310/SDG.2006.v11.n1.a9