Nous considérons les classes conformes auto-duales sur introduites par LeBrun. Elles dépendent du choix de points dans l’espace hyperbolique de dimension 3, appelés points de monopôle. Nous étudions les limites de diverses métriques de courbure scalaire constante dans ces classes conformes lorsque ces points se rapprochent ou tendent vers le bord de l’espace hyperbolique. Il existe une relation étroite avec le problème de Yamabe sur les orbifolds qui n’admet pas toujours de solution (contrairement au cas des variétés compactes). En particulier, nous montrons qu’il n’existe pas de métrique d’orbifold de courbure scalaire constante dans la classe conforme d’un espace ALE hyperkählérien conformément compact en dimension 4.
We consider the self-dual conformal classes on discovered by LeBrun. These depend upon a choice of points in hyperbolic -space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the conformal class of a conformally compactified non-flat hyperkähler ALE space in dimension four.
@article{AIF_2010__60_7_2503_0, author = {Viaclovsky, Jeff A.}, title = {Monopole metrics and the orbifold Yamabe problem}, journal = {Annales de l'Institut Fourier}, volume = {60}, year = {2010}, pages = {2503-2543}, doi = {10.5802/aif.2617}, zbl = {1227.53060}, mrnumber = {2866998}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_2010__60_7_2503_0} }
Viaclovsky, Jeff A. Monopole metrics and the orbifold Yamabe problem. Annales de l'Institut Fourier, Tome 60 (2010) pp. 2503-2543. doi : 10.5802/aif.2617. http://gdmltest.u-ga.fr/item/AIF_2010__60_7_2503_0/
[1] Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl., Tome 4 (1994) no. 3, pp. 239-258 | Article | MR 1299397 | Zbl 0810.53030
[2] Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature, Pacific J. Math., Tome 175 (1996) no. 2, pp. 307-335 | MR 1432834 | Zbl 0881.53036
[3] Computations of the orbifold Yamabe invariant (2010) (arXiv.org:1009.3576)
[4] Yamabe metrics on cylindrical manifolds, Geom. Funct. Anal., Tome 13 (2003) no. 2, pp. 259-333 | Article | MR 1982146 | Zbl 1161.53344
[5] The Yamabe invariants of orbifolds and cylindrical manifolds, and -harmonic spinors, J. Reine Angew. Math., Tome 574 (2004), pp. 121-146 | Article | MR 2099112 | Zbl 1055.53035
[6] Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc., Tome 2 (1989) no. 3, pp. 455-490 | Article | MR 999661 | Zbl 0694.53045
[7] Orbifold compactness for spaces of Riemannian metrics and applications, Math. Ann., Tome 331 (2005) no. 4, pp. 739-778 | Article | MR 2148795 | Zbl 1071.53025
[8] Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys., Tome 125 (1989) no. 4, pp. 637-642 | Article | MR 1024931 | Zbl 0734.53051
[9] Nonlinear analysis on manifolds. Monge-Ampère equations, Springer-Verlag, New York, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 252 (1982) | MR 681859 | Zbl 0512.53044
[10] Bubbling out of Einstein manifolds, Tohoku Math. J. (2), Tome 42 (1990) no. 2, pp. 205-216 | Article | MR 1053949 | Zbl 0719.53025
[11] The mass of an asymptotically flat manifold, Comm. Pure Appl. Math., Tome 39 (1986) no. 5, pp. 661-693 | Article | MR 849427 | Zbl 0598.53045
[12] Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., Tome 42 (1989) no. 3, pp. 271-297 | Article | MR 982351 | Zbl 0702.35085
[13] An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2), Tome 155 (2002) no. 3, pp. 709-787 | Article | MR 1923964 | Zbl 1031.53062
[14] On a conformal gap and finiteness theorem for a class of four-manifolds, Geom. Funct. Anal., Tome 17 (2007) no. 2, pp. 404-434 | Article | MR 2322490 | Zbl 1124.53020
[15] On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc., Tome 21 (2008) no. 4, pp. 1137-1168 | Article | MR 2425183 | Zbl pre05817582
[16] Moduli spaces of critical riemannian metrics with norm curvature bounds (2007) (to appear in Advances in Mathematics) | Zbl 1205.53074
[17] Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity, Tome 2 (1989) no. 2, pp. 197-239 | Article | MR 994091 | Zbl 0671.53029
[18] Self-dual conformal structures on , J. Differential Geom., Tome 33 (1991) no. 2, pp. 551-573 | MR 1094469 | Zbl 0736.53046
[19] Gravitational multi-instantons, Physics Letters B, Tome 78 (1978) no. 4, pp. 430 -432 | Article
[20] A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom., Tome 63 (2003) no. 1, pp. 131-154 | MR 2015262 | Zbl 1070.53018
[21] From the Yamabe problem to the equivariant Yamabe problem, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Soc. Math. France, Paris (Sémin. Congr.) Tome 1 (1996), pp. 377-402 (Joint work with M. Vaugon) | MR 1427765 | Zbl 0880.53035
[22] Polygons and gravitons, Math. Proc. Cambridge Philos. Soc., Tome 85 (1979) no. 3, pp. 465-476 | Article | MR 520463 | Zbl 0405.53016
[23] Einstein metrics and the eta-invariant, Boll. Un. Mat. Ital. B (7), Tome 11 (1997) no. 2, suppl., pp. 95-105 | MR 1456253 | Zbl 0973.53519
[24] Degenerations of LeBrun twistor spaces (2010) (to appear in Communications in Mathematical Physics)
[25] Conformal symmetries of self-dual hyperbolic monopole metrics (2009) (arXiv.org:0902.2019)
[26] Explicit construction of self-dual -manifolds, Duke Math. J., Tome 77 (1995) no. 3, pp. 519-552 | Article | MR 1324633 | Zbl 0855.57028
[27] Constant scalar curvature metrics on connected sums, Int. J. Math. Math. Sci. (2003) no. 7, pp. 405-450 | Article | MR 1961016 | Zbl 1026.53019
[28] Scalar curvature of a metric with unit volume, Math. Ann., Tome 279 (1987) no. 2, pp. 253-265 | Article | MR 919505 | Zbl 0611.53037
[29] The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom., Tome 29 (1989) no. 3, pp. 665-683 | MR 992334 | Zbl 0671.53045
[30] Conformal transformations between Einstein spaces, Conformal geometry (Bonn, 1985/1986), Vieweg, Braunschweig (Aspects Math., E12) (1988), pp. 105-146 | MR 979791 | Zbl 0667.53039
[31] Counter-examples to the generalized positive action conjecture, Comm. Math. Phys., Tome 118 (1988) no. 4, pp. 591-596 | Article | MR 962489 | Zbl 0659.53050
[32] Explicit self-dual metrics on , J. Differential Geom., Tome 34 (1991) no. 1, pp. 223-253 | MR 1114461 | Zbl 0725.53067
[33] Self-dual manifolds with positive Ricci curvature, Math. Z., Tome 224 (1997) no. 1, pp. 49-63 | Article | MR 1427703 | Zbl 0868.53032
[34] The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), Tome 17 (1987) no. 1, pp. 37-91 | Article | MR 888880 | Zbl 0633.53062
[35] Connected sum constructions for constant scalar curvature metrics, Topol. Methods Nonlinear Anal., Tome 6 (1995) no. 2, pp. 207-233 | MR 1399537 | Zbl 0866.58069
[36] Self-duality of ALE Ricci-flat -manifolds and positive mass theorem, Recent topics in differential and analytic geometry, Academic Press, Boston, MA (Adv. Stud. Pure Math.) Tome 18 (1990), pp. 385-396 | MR 1145266 | Zbl 0744.53025
[37] A convergence theorem for Einstein metrics and the ALE spaces [ MR1193019 (93k:53044)], Selected papers on number theory, algebraic geometry, and differential geometry, Amer. Math. Soc., Providence, RI (Amer. Math. Soc. Transl. Ser. 2) Tome 160 (1994), pp. 79-94 | MR 1308542
[38] The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, Tome 6 (1971/72), pp. 247-258 | MR 303464 | Zbl 0236.53042
[39] Compact self-dual manifolds with positive scalar curvature, J. Differential Geom., Tome 24 (1986) no. 1, pp. 97-132 | MR 857378 | Zbl 0583.53054
[40] Foundations of hyperbolic manifolds, Springer, New York, Graduate Texts in Mathematics, Tome 149 (2006) | MR 2249478 | Zbl 1106.51009
[41] Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Tome 20 (1984) no. 2, pp. 479-495 | MR 788292 | Zbl 0576.53028
[42] On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Longman Sci. Tech., Harlow (Pitman Monogr. Surveys Pure Appl. Math.) Tome 52 (1991), pp. 311-320 | MR 1173050 | Zbl 0733.53021
[43] Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., Tome 117 (1965), pp. 251-275 | Article | MR 174022 | Zbl 0136.17701
[44] On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., Tome 101 (1990) no. 1, pp. 101-172 | Article | MR 1055713 | Zbl 0716.32019
[45] Bach-flat asymptotically locally Euclidean metrics, Invent. Math., Tome 160 (2005) no. 2, pp. 357-415 | Article | MR 2138071 | Zbl 1085.53030
[46] Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math., Tome 196 (2005) no. 2, pp. 346-372 | Article | MR 2166311 | Zbl pre02213018
[47] Volume growth, curvature decay, and critical metrics, Comment. Math. Helv., Tome 83 (2008) no. 4, pp. 889-911 | Article | MR 2442967 | Zbl 1154.53024