[La masse de Gauss–Bonnet–Chern sur des variétés asymptotiquement hyperboliques]
En utilisant la courbure de Gauss–Bonnet, on introduit une nouvelle masse dʼordre supérieur – la masse de Gauss–Bonnet–Chern –, sur des variétés asymptotiquement hyperboliques. On montre quʼil sʼagit dʼun invariant géométrique. On démontre également le théorème de masse positive sur des graphes sur lʼespace hyperbolique et des inégalités dʼAlexandrov–Fenchel à poids dans pour toute hypersurface convexe de type horosphérique. Ainsi, on obtient une inégalité de type Penrose optimale pour cette masse sur toute variété asymptotiquement hyperbolique qui est graphe sur avec un horizon au bord, à condition que la condition dʼénergie dominante soit satisfaite.
By using the Gauss–Bonnet curvature, we introduce a higher-order mass, the Gauss–Bonnet–Chern mass, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs. Then, we prove the weighted Alexandrov–Fenchel inequalities in the hyperbolic space for any horospherical convex hypersurface Σ. As an application, we obtain an optimal Penrose-type inequality for this new mass for asymptotically hyperbolic graphs with a horizon type boundary Σ, provided that a dominant energy condition holds.
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@article{CRMATH_2014__352_2_147_0,
author = {Ge, Yuxin and Wang, Guofang and Wu, Jie},
title = {The {GBC} mass for asymptotically hyperbolic manifolds},
journal = {Comptes Rendus. Math\'ematique},
pages = {147--151},
publisher = {Elsevier},
volume = {352},
number = {2},
year = {2014},
doi = {10.1016/j.crma.2013.11.019},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2013.11.019/}
}
TY - JOUR AU - Ge, Yuxin AU - Wang, Guofang AU - Wu, Jie TI - The GBC mass for asymptotically hyperbolic manifolds JO - Comptes Rendus. Mathématique PY - 2014 SP - 147 EP - 151 VL - 352 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2013.11.019/ DO - 10.1016/j.crma.2013.11.019 LA - en ID - CRMATH_2014__352_2_147_0 ER -
%0 Journal Article %A Ge, Yuxin %A Wang, Guofang %A Wu, Jie %T The GBC mass for asymptotically hyperbolic manifolds %J Comptes Rendus. Mathématique %D 2014 %P 147-151 %V 352 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2013.11.019/ %R 10.1016/j.crma.2013.11.019 %G en %F CRMATH_2014__352_2_147_0
Ge, Yuxin; Wang, Guofang; Wu, Jie. The GBC mass for asymptotically hyperbolic manifolds. Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 147-151. doi : 10.1016/j.crma.2013.11.019. http://www.numdam.org/articles/10.1016/j.crma.2013.11.019/
[1] Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differ. Geom., Volume 59 (2001), pp. 177-267
[2] On the Riemannian Penrose inequality in dimensions less than eight, Duke Math. J., Volume 148 (2009), pp. 81-106
[3] The mass of asymptotically hyperbolic Riemannian manifolds, Pac. J. Math., Volume 212 (2003), pp. 231-264
[4] Penrose type inequalities for asymptotically hyperbolic graphs, Ann. Inst. Henri Poincaré, Volume 14 (2013) no. 5, pp. 1135-1168
[5] An Alexandrov–Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality | arXiv
[6] A new mass for asymptotically flat manifolds | arXiv
[7] The Gauss–Bonnet–Chern mass of conformally flat manifolds (to appear in Int. Math. Res. Not) | arXiv
[8] Y. Ge, G. Wang, J. Wu, The GBC mass for asymptotically hyperbolic manifolds, preprint.
[9] , Eur. Math. Soc., Zurich (2005), pp. 103-121
[10] The equality case of the Penrose inequality for asymptotically flat graphs | arXiv
[11] The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differ. Geom., Volume 59 (2001), pp. 353-437
[12] The graph cases of the Riemannian positive mass and Penrose inequality in all dimensions | arXiv
[13] The Einstein tensor and its generalizations, J. Math. Phys., Volume 12 (1971), pp. 498-501
[14] Geometric invariance of mass-like asymptotic invariants, J. Math. Phys., Volume 52 (2011) no. 5, p. 052504
[15] On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., Volume 65 (1979), pp. 45-76
[16] Mass for asymptotically hyperbolic manifolds, J. Differ. Geom., Volume 57 (2001), pp. 273-299
[17] A new proof of the positive energy theorem, Commun. Math. Phys., Volume 80 (1981), pp. 381-402
[18] Lectures on Chern–Weil Theory and Witten Deformations, Nankai Tracts in Mathematics, vol. 4, World Scientific Publishing Co., Inc., River Edge, NJ, 2001
[19] A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds. I, Commun. Math. Phys., Volume 249 (2004), pp. 529-548
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☆ This project is partly supported by SFB/TR71 “Geometric partial differential equations” of DFG.
