Monopole metrics and the orbifold Yamabe problem
Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2503-2543.

We consider the self-dual conformal classes on n#ℂℙ 2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-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.

Nous considérons les classes conformes auto-duales sur n#ℂℙ 2 introduites par LeBrun. Elles dépendent du choix de n 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.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2617
Classification: 53C21,  53C25
Keywords: Monopole Metrics, Orbifold Yamabe Problem
Viaclovsky, Jeff A. 1

1 University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706 (USA)
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Viaclovsky, Jeff A. Monopole metrics and the orbifold Yamabe problem. Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2503-2543. doi : 10.5802/aif.2617. https://aif.centre-mersenne.org/articles/10.5802/aif.2617/

[1] Akutagawa, Kazuo 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] Akutagawa, Kazuo 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] Akutagawa, Kazuo Computations of the orbifold Yamabe invariant (2010) (arXiv.org:1009.3576)

[4] Akutagawa, Kazuo; Botvinnik, Boris Yamabe metrics on cylindrical manifolds, Geom. Funct. Anal., Volume 13 (2003) no. 2, pp. 259-333 | DOI | MR | Zbl

[5] Akutagawa, Kazuo; Botvinnik, Boris The Yamabe invariants of orbifolds and cylindrical manifolds, and L 2 -harmonic spinors, J. Reine Angew. Math., Volume 574 (2004), pp. 121-146 | DOI | MR | Zbl

[6] Anderson, Michael T. 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] Anderson, Michael T. Orbifold compactness for spaces of Riemannian metrics and applications, Math. Ann., Volume 331 (2005) no. 4, pp. 739-778 | DOI | MR | Zbl

[8] Anderson, Michael T.; Kronheimer, Peter B.; LeBrun, Claude Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys., Volume 125 (1989) no. 4, pp. 637-642 | DOI | MR | Zbl

[9] Aubin, Thierry Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 252, Springer-Verlag, New York, 1982 | MR | Zbl

[10] Bando, Shigetoshi Bubbling out of Einstein manifolds, Tohoku Math. J. (2), Volume 42 (1990) no. 2, pp. 205-216 | DOI | MR | Zbl

[11] Bartnik, Robert The mass of an asymptotically flat manifold, Comm. Pure Appl. Math., Volume 39 (1986) no. 5, pp. 661-693 | DOI | MR | Zbl

[12] Caffarelli, Luis A.; Gidas, Basilis; Spruck, Joel 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] Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C. 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] Chang, Sun-Yung A.; Qing, Jie; Yang, Paul 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] Chen, Xiuxiong; Lebrun, Claude; Weber, Brian On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc., Volume 21 (2008) no. 4, pp. 1137-1168 | DOI | MR | Zbl

[16] Chen, Xiuxiong; Weber, Brian Moduli spaces of critical riemannian metrics with L n/2 norm curvature bounds (2007) (to appear in Advances in Mathematics) | Zbl

[17] Donaldson, S.; Friedman, R. Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity, Volume 2 (1989) no. 2, pp. 197-239 | DOI | MR | Zbl

[18] Floer, Andreas Self-dual conformal structures on lCP 2 , J. Differential Geom., Volume 33 (1991) no. 2, pp. 551-573 | MR | Zbl

[19] Gibbons, G.W.; Hawking, S.W. Gravitational multi-instantons, Physics Letters B, Volume 78 (1978) no. 4, pp. 430 -432 | DOI

[20] Gursky, Matthew J.; Viaclovsky, Jeff A. 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] Hebey, Emmanuel 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] Hitchin, N. J. Polygons and gravitons, Math. Proc. Cambridge Philos. Soc., Volume 85 (1979) no. 3, pp. 465-476 | DOI | MR | Zbl

[23] Hitchin, N. J. Einstein metrics and the eta-invariant, Boll. Un. Mat. Ital. B (7), Volume 11 (1997) no. 2, suppl., pp. 95-105 | MR | Zbl

[24] Honda, Nobuhiro Degenerations of LeBrun twistor spaces (2010) (to appear in Communications in Mathematical Physics)

[25] Honda, Nobuhiro; Viaclovsky, Jeff Conformal symmetries of self-dual hyperbolic monopole metrics (2009) (arXiv.org:0902.2019)

[26] Joyce, Dominic Explicit construction of self-dual 4-manifolds, Duke Math. J., Volume 77 (1995) no. 3, pp. 519-552 | DOI | MR | Zbl

[27] Joyce, Dominic Constant scalar curvature metrics on connected sums, Int. J. Math. Math. Sci. (2003) no. 7, pp. 405-450 | DOI | MR | Zbl

[28] Kobayashi, Osamu Scalar curvature of a metric with unit volume, Math. Ann., Volume 279 (1987) no. 2, pp. 253-265 | DOI | MR | Zbl

[29] Kronheimer, P. B. The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom., Volume 29 (1989) no. 3, pp. 665-683 | MR | Zbl

[30] Kühnel, Wolfgang Conformal transformations between Einstein spaces, Conformal geometry (Bonn, 1985/1986) (Aspects Math., E12), Vieweg, Braunschweig, 1988, pp. 105-146 | MR | Zbl

[31] LeBrun, Claude Counter-examples to the generalized positive action conjecture, Comm. Math. Phys., Volume 118 (1988) no. 4, pp. 591-596 | DOI | MR | Zbl

[32] LeBrun, Claude Explicit self-dual metrics on CP 2 ##CP 2 , J. Differential Geom., Volume 34 (1991) no. 1, pp. 223-253 | MR | Zbl

[33] LeBrun, Claude; Nayatani, Shin; Nitta, Takashi Self-dual manifolds with positive Ricci curvature, Math. Z., Volume 224 (1997) no. 1, pp. 49-63 | DOI | MR | Zbl

[34] Lee, John M.; Parker, Thomas H. The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), Volume 17 (1987) no. 1, pp. 37-91 | DOI | MR | Zbl

[35] Mazzeo, Rafe; Pollack, Daniel; Uhlenbeck, Karen Connected sum constructions for constant scalar curvature metrics, Topol. Methods Nonlinear Anal., Volume 6 (1995) no. 2, pp. 207-233 | MR | Zbl

[36] Nakajima, Hiraku Self-duality of ALE Ricci-flat 4-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] Nakajima, Hiraku 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] Obata, Morio The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, Volume 6 (1971/72), pp. 247-258 | MR | Zbl

[39] Poon, Y. Sun Compact self-dual manifolds with positive scalar curvature, J. Differential Geom., Volume 24 (1986) no. 1, pp. 97-132 | MR | Zbl

[40] Ratcliffe, John G. Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, Volume 149, Springer, New York, 2006 | MR | Zbl

[41] Schoen, Richard Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Volume 20 (1984) no. 2, pp. 479-495 | MR | Zbl

[42] Schoen, Richard 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] Tashiro, Yoshihiro Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., Volume 117 (1965), pp. 251-275 | DOI | MR | Zbl

[44] Tian, Gang 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] Tian, Gang; Viaclovsky, Jeff Bach-flat asymptotically locally Euclidean metrics, Invent. Math., Volume 160 (2005) no. 2, pp. 357-415 | DOI | MR | Zbl

[46] Tian, Gang; Viaclovsky, Jeff Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math., Volume 196 (2005) no. 2, pp. 346-372 | DOI | MR | Zbl

[47] Tian, Gang; Viaclovsky, Jeff Volume growth, curvature decay, and critical metrics, Comment. Math. Helv., Volume 83 (2008) no. 4, pp. 889-911 | DOI | MR | Zbl

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