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.
Keywords: Monopole Metrics, Orbifold Yamabe Problem
Mot clés : Métriques de monopôles, Problème de Yamabe sur les Orbifolds
@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}, pages = {2503--2543}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {7}, year = {2010}, doi = {10.5802/aif.2617}, zbl = {1227.53060}, mrnumber = {2866998}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2617/} }
TY - JOUR AU - Viaclovsky, Jeff A. TI - Monopole metrics and the orbifold Yamabe problem JO - Annales de l'Institut Fourier PY - 2010 SP - 2503 EP - 2543 VL - 60 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2617/ DO - 10.5802/aif.2617 LA - en ID - AIF_2010__60_7_2503_0 ER -
%0 Journal Article %A Viaclovsky, Jeff A. %T Monopole metrics and the orbifold Yamabe problem %J Annales de l'Institut Fourier %D 2010 %P 2503-2543 %V 60 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2617/ %R 10.5802/aif.2617 %G en %F AIF_2010__60_7_2503_0
Viaclovsky, Jeff A. Monopole metrics and the orbifold Yamabe problem. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2503-2543. doi : 10.5802/aif.2617. http://www.numdam.org/articles/10.5802/aif.2617/
[1] Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl., Volume 4 (1994) no. 3, pp. 239-258 | DOI | MR | Zbl
[2] Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature, Pacific J. Math., Volume 175 (1996) no. 2, pp. 307-335 | MR | Zbl
[3] Computations of the orbifold Yamabe invariant (2010) (arXiv.org:1009.3576)
[4] Yamabe metrics on cylindrical manifolds, Geom. Funct. Anal., Volume 13 (2003) no. 2, pp. 259-333 | DOI | MR | Zbl
[5] The Yamabe invariants of orbifolds and cylindrical manifolds, and -harmonic spinors, J. Reine Angew. Math., Volume 574 (2004), pp. 121-146 | DOI | MR | Zbl
[6] Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc., Volume 2 (1989) no. 3, pp. 455-490 | DOI | MR | Zbl
[7] Orbifold compactness for spaces of Riemannian metrics and applications, Math. Ann., Volume 331 (2005) no. 4, pp. 739-778 | DOI | MR | Zbl
[8] Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys., Volume 125 (1989) no. 4, pp. 637-642 | DOI | MR | Zbl
[9] Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 252, Springer-Verlag, New York, 1982 | MR | Zbl
[10] Bubbling out of Einstein manifolds, Tohoku Math. J. (2), Volume 42 (1990) no. 2, pp. 205-216 | DOI | MR | Zbl
[11] The mass of an asymptotically flat manifold, Comm. Pure Appl. Math., Volume 39 (1986) no. 5, pp. 661-693 | DOI | MR | Zbl
[12] Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., Volume 42 (1989) no. 3, pp. 271-297 | DOI | MR | Zbl
[13] An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2), Volume 155 (2002) no. 3, pp. 709-787 | DOI | MR | Zbl
[14] On a conformal gap and finiteness theorem for a class of four-manifolds, Geom. Funct. Anal., Volume 17 (2007) no. 2, pp. 404-434 | DOI | MR | Zbl
[15] On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc., Volume 21 (2008) no. 4, pp. 1137-1168 | DOI | MR
[16] Moduli spaces of critical riemannian metrics with norm curvature bounds (2007) (to appear in Advances in Mathematics) | Zbl
[17] Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity, Volume 2 (1989) no. 2, pp. 197-239 | DOI | MR | Zbl
[18] Self-dual conformal structures on , J. Differential Geom., Volume 33 (1991) no. 2, pp. 551-573 | MR | Zbl
[19] Gravitational multi-instantons, Physics Letters B, Volume 78 (1978) no. 4, pp. 430 -432 | DOI
[20] A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom., Volume 63 (2003) no. 1, pp. 131-154 | MR | Zbl
[21] From the Yamabe problem to the equivariant Yamabe problem, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) (Sémin. Congr.), Volume 1, Soc. Math. France, Paris, 1996, pp. 377-402 (Joint work with M. Vaugon) | MR | Zbl
[22] Polygons and gravitons, Math. Proc. Cambridge Philos. Soc., Volume 85 (1979) no. 3, pp. 465-476 | DOI | MR | Zbl
[23] Einstein metrics and the eta-invariant, Boll. Un. Mat. Ital. B (7), Volume 11 (1997) no. 2, suppl., pp. 95-105 | MR | Zbl
[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., Volume 77 (1995) no. 3, pp. 519-552 | DOI | MR | Zbl
[27] Constant scalar curvature metrics on connected sums, Int. J. Math. Math. Sci. (2003) no. 7, pp. 405-450 | DOI | MR | Zbl
[28] Scalar curvature of a metric with unit volume, Math. Ann., Volume 279 (1987) no. 2, pp. 253-265 | DOI | MR | Zbl
[29] The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom., Volume 29 (1989) no. 3, pp. 665-683 | MR | Zbl
[30] Conformal transformations between Einstein spaces, Conformal geometry (Bonn, 1985/1986) (Aspects Math., E12), Vieweg, Braunschweig, 1988, pp. 105-146 | MR | Zbl
[31] Counter-examples to the generalized positive action conjecture, Comm. Math. Phys., Volume 118 (1988) no. 4, pp. 591-596 | DOI | MR | Zbl
[32] Explicit self-dual metrics on , J. Differential Geom., Volume 34 (1991) no. 1, pp. 223-253 | MR | Zbl
[33] Self-dual manifolds with positive Ricci curvature, Math. Z., Volume 224 (1997) no. 1, pp. 49-63 | DOI | MR | Zbl
[34] The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), Volume 17 (1987) no. 1, pp. 37-91 | DOI | MR | Zbl
[35] Connected sum constructions for constant scalar curvature metrics, Topol. Methods Nonlinear Anal., Volume 6 (1995) no. 2, pp. 207-233 | MR | Zbl
[36] Self-duality of ALE Ricci-flat -manifolds and positive mass theorem, Recent topics in differential and analytic geometry (Adv. Stud. Pure Math.), Volume 18, Academic Press, Boston, MA, 1990, pp. 385-396 | MR | Zbl
[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. Transl. Ser. 2), Volume 160, Amer. Math. Soc., Providence, RI, 1994, pp. 79-94 | MR
[38] The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, Volume 6 (1971/72), pp. 247-258 | MR | Zbl
[39] Compact self-dual manifolds with positive scalar curvature, J. Differential Geom., Volume 24 (1986) no. 1, pp. 97-132 | MR | Zbl
[40] Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149, Springer, New York, 2006 | MR | Zbl
[41] Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Volume 20 (1984) no. 2, pp. 479-495 | MR | Zbl
[42] On the number of constant scalar curvature metrics in a conformal class, Differential geometry (Pitman Monogr. Surveys Pure Appl. Math.), Volume 52, Longman Sci. Tech., Harlow, 1991, pp. 311-320 | MR | Zbl
[43] Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., Volume 117 (1965), pp. 251-275 | DOI | MR | Zbl
[44] On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., Volume 101 (1990) no. 1, pp. 101-172 | DOI | MR | Zbl
[45] Bach-flat asymptotically locally Euclidean metrics, Invent. Math., Volume 160 (2005) no. 2, pp. 357-415 | DOI | MR | Zbl
[46] Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math., Volume 196 (2005) no. 2, pp. 346-372 | DOI | MR
[47] Volume growth, curvature decay, and critical metrics, Comment. Math. Helv., Volume 83 (2008) no. 4, pp. 889-911 | DOI | MR | Zbl
Cité par Sources :