A mass for asymptotically complex hyperbolic manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, p. 875-902

We prove a positive-mass theorem for complete Kähler manifolds that are asymptotic to the complex hyperbolic space.

Published online : 2018-06-21
Classification:  53C24,  53C27,  53C55,  58J60
@article{ASNSP_2012_5_11_4_875_0,
     author = {Maerten, Daniel and Minerbe, Vincent},
     title = {A mass for asymptotically complex hyperbolic manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {4},
     year = {2012},
     pages = {875-902},
     zbl = {1269.53041},
     mrnumber = {3060704},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_4_875_0}
}
Maerten, Daniel; Minerbe, Vincent. A mass for asymptotically complex hyperbolic manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 875-902. http://www.numdam.org/item/ASNSP_2012_5_11_4_875_0/

[1] L. Andersson and M. Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), 1–27. | MR 1616570 | Zbl 0946.53021

[2] V. Apostolov, D. M. J. Calderbank and P. Gauduchon, Hamiltonian 2-forms in Kähler geometry. I. General theory, J. Differential Geom. 73 (2006), 359–412. | MR 2228318 | Zbl 1101.53041

[3] R. Arnowitt, S. Deser and C. W. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. (2) 122 (1961), 997–1006. | MR 127946 | Zbl 0094.23003

[4] R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661–693. | MR 849427 | Zbl 0598.53045

[5] H. Baum, Complete Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom. 7 (1989), 205–226. | MR 1039119 | Zbl 0694.53043

[6] O. Biquard, “Asymptotically Symmetric Einstein Metrics”, Translated from the 2000 French original by Stephen S. Wilson. SMF/AMS Texts and Monographs, 13. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2006. | MR 2260400 | Zbl 1112.53001

[7] H. Boualem and M. Herzlich, Rigidity at infinity for even-dimensional asymptotically complex hyperbolic spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), 461–469. | Numdam | MR 1991147 | Zbl 1170.53308

[8] P. T. Chruściel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), 231–264. | MR 2038048 | Zbl 1056.53025

[9] X. Dai, A positive-mass theorem for spaces with asymptotic SUSY compactification, Comm. Math. Phys. 244 (2004), 335–345. | MR 2031034 | Zbl 1075.83013

[10] S. Gallot, Équations différentielles caractéristiques de la sphère, Ann. Sci. École Norm. Sup. 12 (1979), 235–267. | Numdam | MR 543217 | Zbl 0412.58009

[11] M. Herzlich, Scalar curvature and rigidity of odd-dimensional complex hyperbolic spaces, Math. Ann. 312 (1998), 641–657. | MR 1660251 | Zbl 0946.53022

[12] A. D. Hwang and M. A. Singer A momentum construction for circle-invariant Kähler metrics, Trans. Amer. Math. Soc. 354 (2002), 2285–2325. | MR 1885653 | Zbl 0987.53032

[13] K. D. Kirchberg, Killing spinors on Kähler manifolds, Ann. Global Anal. Geom. 11 (1993), 141–164. | MR 1225435 | Zbl 0810.53033

[14] S. Kobayashi and K. Nomizu, “Foundations of Differential Geometry”, Vol. II. Reprint of the 1969 original, Wiley Classics Library, New York, 1996. | MR 1393941 | Zbl 0119.37502

[15] H. Blaine Lawson and Jr, Marie-Louise Michelsohn, “Spin Geometry”, Princeton Mathematical Series, 38, Princeton University Press, Princeton, NJ, 1989. | MR 1031992 | Zbl 0688.57001

[16] J. M. Lee and T. H. Parker, The Yamabe problem, Bull Amer. Math. Soc. (N.S.) 17 (1987), 37–91. | MR 888880 | Zbl 0633.53062

[17] J. Lohkamp, The higher dimensional positive mass theorem I, arXiv:math/0608795v1.

[18] V. Minerbe, A mass for ALF manifolds, Comm. Math. Phys. 289 (2009), 925–955. | MR 2511656 | Zbl 1183.53066

[19] M. Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989), 527–539. | MR 1027758 | Zbl 0686.53038

[20] A. Moroianu, La première valeur propre de l’opérateur de Dirac sur les variétés kählériennes compactes, Comm. Math. Phys. 169 (1995), 373–384. | MR 1329200 | Zbl 0832.53054

[21] M. Obata, Riemannian manifolds admitting a solution of a certain system of differential equations, In: “Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965)”, Nippon Hyoronsha, Tokyo, 1966, 101–114. | MR 216430 | Zbl 0144.20903

[22] R. Schoen and S. T. Yau, On the proof of the positive-mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45–76. | MR 526976 | Zbl 0405.53045

[23] R. Schoen and S. T. Yau, Proof of the positive-mass theorem. II, Comm. Math. Phys. 79 (1981), 231–260. | MR 612249 | Zbl 0494.53028

[24] Y. Shi and L.-F. Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), 79–125. | MR 1987378 | Zbl 1071.53018

[25] E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381–402. | MR 626707 | Zbl 1051.83532