Nous étudions la géométrie à l'infini des instantons gravitationnels, i.e. des variétés hyperkählériennes, asymptotiquement plates et de dimension quatre. En particulier, nous prouvons que les instantons gravitationnels dont la croissance de volume est cubique sont asymptotiques à une fibration en cercles au-dessus d'un espace euclidien à trois dimensions, avec des fibres de longueur asymptotiquement constante ; autrement dit, ils sont ALF (asymptotically locally flat).
We investigate the geometry at infinity of the so-called “gravitational instantons”, i.e. asymptotically flat hyperkähler four-manifolds, in relation with their volume growth. In particular, we prove that gravitational instantons with cubic volume growth are ALF, namely asymptotic to a circle fibration over a Euclidean three-space, with fibers of asymptotically constant length.
@article{ASENS_2010_4_43_6_883_0,
author = {Minerbe, Vincent},
title = {On the asymptotic geometry of gravitational instantons},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
volume = {43},
year = {2010},
pages = {883-924},
doi = {10.24033/asens.2135},
mrnumber = {2778451},
zbl = {1215.53043},
language = {en},
url = {http://dml.mathdoc.fr/item/ASENS_2010_4_43_6_883_0}
}
Minerbe, Vincent. On the asymptotic geometry of gravitational instantons. Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010) pp. 883-924. doi : 10.24033/asens.2135. http://gdmltest.u-ga.fr/item/ASENS_2010_4_43_6_883_0/
[1] , Lower curvature bounds, Toponogov's theorem, and bounded topology. II, Ann. Sci. École Norm. Sup. 20 (1987), 475-502. | Numdam | MR 925724 | Zbl 0651.53031
[2] , & , On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), 313-349. | MR 1001844 | Zbl 0682.53045
[3] , Einstein manifolds, Ergebnisse Math. Grenzg. 10, Springer, 1987. | MR 867684 | Zbl 0613.53001
[4] & , Gromov's almost flat manifolds, Astérisque 81, Soc. Math. France, 1981. | MR 619537 | Zbl 0459.53031
[5] , & , Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), 327-372. | MR 1126118 | Zbl 0758.53022
[6] & , Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990), 269-298. | MR 1064875 | Zbl 0727.53043
[7] , & , Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15-53. | MR 658471 | Zbl 0493.53035
[8] & , Gravitational instantons of type , Comm. Math. Phys. 260 (2005), 299-317. | MR 2177322 | Zbl 1085.53038
[9] & , gravitational instantons and Nahm equations, Adv. Theor. Math. Phys. 2 (1998), 1287-1306. | MR 1693628 | Zbl 0945.58017
[10] & , Singular monopoles and gravitational instantons, Comm. Math. Phys. 203 (1999), 713-728. | MR 1700937 | Zbl 0960.83007
[11] & , Hyper-Kähler metrics from periodic monopoles, Phys. Rev. D 65 (2002), 084015, 10. | MR 1899201
[12] , Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. 13 (1980), 419-435. | Numdam | MR 608287 | Zbl 0465.53032
[13] & , On Yang-Mills instantons over multi-centered gravitational instantons, Comm. Math. Phys. 235 (2003), 275-288. | MR 1969729 | Zbl 1028.81039
[14] & , Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons, Comm. Math. Phys. 280 (2008), 285-313. | MR 2395472 | Zbl 1144.53035
[15] , Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987), 139-156. | MR 873459 | Zbl 0606.53027
[16] , Structures métriques pour les variétés riemanniennes, Textes Mathématiques 1, CEDIC, 1981. | MR 682063 | Zbl 0509.53034
[17] , Metric structures for Riemannian and non-Riemannian spaces, Progress in Math. 152, Birkhäuser, 1999. | MR 1699320 | Zbl 0953.53002
[18] , & , Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004), 485-548. | MR 2057017 | Zbl 1062.58002
[19] , Gravitational instantons, Phys. Lett. A 60 (1977), 81-83. | MR 465052
[20] , -cohomology of hyperkähler quotients, Comm. Math. Phys. 211 (2000), 153-165. | MR 1757010 | Zbl 0955.58019
[21] , A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. 21 (1988), 593-622. | Numdam | MR 982335 | Zbl 0662.53032
[22] , Schranken für die Christoffelsymbole, Manuscripta Math. 19 (1976), 261-273. | MR 433351 | Zbl 0332.53025
[23] , The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665-683. | MR 992334 | Zbl 0671.53045
[24] , A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), 685-697. | MR 992335 | Zbl 0671.53046
[25] , Complete Ricci-flat Kähler metrics on need not be flat, in Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math. 52, Amer. Math. Soc., 1991, 297-304. | MR 1128554 | Zbl 0739.53053
[26] & , Green's functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), 277-318. | MR 1331970 | Zbl 0827.53033
[27] & , Manifolds with quadratic curvature decay and slow volume growth, Ann. Sci. École Norm. Sup. 33 (2000), 275-290. | Numdam | MR 1755117 | Zbl 0996.53026
[28] , Weighted Sobolev inequalities and Ricci flat manifolds, Geom. Funct. Anal. 18 (2009), 1696-1749. | MR 2481740 | Zbl 1166.53028
[29] , A note on Poincaré, Sobolev, and Harnack inequalities, Int. Math. Res. Not. 1992 (1992), 27-38. | MR 1150597 | Zbl 0769.58054
[30] & , Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math. 196 (2005), 346-372. | MR 2166311 | Zbl 1252.53045
[31] , Asymptotically flat -manifolds, Differential Geom. Appl. 6 (1996), 271-274. | MR 1408311 | Zbl 0856.53031