On the asymptotic geometry of gravitational instantons
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 6, pp. 883-924.

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.

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).

DOI: 10.24033/asens.2135
Classification: 53C20, 53C21, 53C23, 53C26, 53C29
Keywords: gravitational instantons, hyperkähler manifolds, asymptotically flat manifolds
Mot clés : instantons gravitationnels, variétés hyperkähleriennes, variétés asymptotiquement plates
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Minerbe, Vincent. On the asymptotic geometry of gravitational instantons. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 6, pp. 883-924. doi : 10.24033/asens.2135. http://archive.numdam.org/articles/10.24033/asens.2135/

[1] U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology. II, Ann. Sci. École Norm. Sup. 20 (1987), 475-502. | Numdam | MR | Zbl

[2] S. Bando, A. Kasue & H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), 313-349. | MR | Zbl

[3] A. L. Besse, Einstein manifolds, Ergebnisse Math. Grenzg. 10, Springer, 1987. | MR | Zbl

[4] P. Buser & H. Karcher, Gromov's almost flat manifolds, Astérisque 81, Soc. Math. France, 1981. | Numdam | MR | Zbl

[5] J. Cheeger, K. Fukaya & M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), 327-372. | MR | Zbl

[6] J. Cheeger & M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990), 269-298. | MR | Zbl

[7] J. Cheeger, M. Gromov & M. Taylor, 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 | Zbl

[8] S. A. Cherkis & N. J. Hitchin, Gravitational instantons of type D k , Comm. Math. Phys. 260 (2005), 299-317. | MR | Zbl

[9] S. A. Cherkis & A. Kapustin, D k gravitational instantons and Nahm equations, Adv. Theor. Math. Phys. 2 (1998), 1287-1306. | MR | Zbl

[10] S. A. Cherkis & A. Kapustin, Singular monopoles and gravitational instantons, Comm. Math. Phys. 203 (1999), 713-728. | MR | Zbl

[11] S. A. Cherkis & A. Kapustin, Hyper-Kähler metrics from periodic monopoles, Phys. Rev. D 65 (2002), 084015, 10. | MR

[12] C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. 13 (1980), 419-435. | Numdam | MR | Zbl

[13] G. Etesi & T. Hausel, On Yang-Mills instantons over multi-centered gravitational instantons, Comm. Math. Phys. 235 (2003), 275-288. | MR | Zbl

[14] G. Etesi & M. Jardim, Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons, Comm. Math. Phys. 280 (2008), 285-313. | MR | Zbl

[15] K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987), 139-156. | MR | Zbl

[16] M. Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques 1, CEDIC, 1981. | MR | Zbl

[17] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Math. 152, Birkhäuser, 1999. | MR | Zbl

[18] T. Hausel, E. Hunsicker & R. Mazzeo, Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004), 485-548. | MR | Zbl

[19] S. W. Hawking, Gravitational instantons, Phys. Lett. A 60 (1977), 81-83. | MR

[20] N. J. Hitchin, L 2 -cohomology of hyperkähler quotients, Comm. Math. Phys. 211 (2000), 153-165. | MR | Zbl

[21] A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. 21 (1988), 593-622. | Numdam | MR | Zbl

[22] H. Kaul, Schranken für die Christoffelsymbole, Manuscripta Math. 19 (1976), 261-273. | MR | Zbl

[23] P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665-683. | MR | Zbl

[24] P. B. Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), 685-697. | MR | Zbl

[25] C. Lebrun, Complete Ricci-flat Kähler metrics on 𝐂 n 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 | Zbl

[26] P. Li & L.-F. Tam, Green's functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), 277-318. | MR | Zbl

[27] J. Lott & Z. Shen, Manifolds with quadratic curvature decay and slow volume growth, Ann. Sci. École Norm. Sup. 33 (2000), 275-290. | Numdam | MR | Zbl

[28] V. Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds, Geom. Funct. Anal. 18 (2009), 1696-1749. | MR | Zbl

[29] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Int. Math. Res. Not. 1992 (1992), 27-38. | MR | Zbl

[30] G. Tian & J. Viaclovsky, Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math. 196 (2005), 346-372. | MR | Zbl

[31] S. Unnebrink, Asymptotically flat 4-manifolds, Differential Geom. Appl. 6 (1996), 271-274. | MR | Zbl

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