×

Moduli spaces of critical Riemannian metrics in dimension four. (English) Zbl 1252.53045

Summary: We obtain a compactness result for various classes of Riemannian metrics in dimension four; in particular our method applies to anti-self-dual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric non-collapsing assumptions, the moduli space can be compactified by adding metrics with orbifold-like singularities. Similar results were obtained for Einstein metrics in [M. T. Anderson, J. Am. Math. Soc. 2, 455-490 (1989; Zbl 0694.53045)], [S. Bando, A. Kasue and H. Nakajima, Invent. Math. 97, No. 2, 313–349 (1989; Zbl 0682.53045)], and [the first author, Invent. Math. 101, No. 1, 101–172 (1990; Zbl 0716.32019)], but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58D27 Moduli problems for differential geometric structures
58E11 Critical metrics

References:

[1] Akutagawa, K., Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl., 4, 3, 239-258 (1994), MR 95i:58046 · Zbl 0810.53030
[2] M.T. Anderson, Orbifold compactness for spaces of Riemannian metrics and applications, preprint, arXiv:math.DG/0312111.; M.T. Anderson, Orbifold compactness for spaces of Riemannian metrics and applications, preprint, arXiv:math.DG/0312111.
[3] Anderson, M. T., Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc., 2, 3, 455-490 (1989), MR 90g:53052 · Zbl 0694.53045
[4] Apostolov, V.; Calderbank, D. M.J.; Gauduchon, P., The geometry of weakly self-dual Kähler surfaces, Compositio Math., 135, 3, 279-322 (2003), MR 1 956 815 · Zbl 1031.53045
[5] Atiyah, M. F.; Hitchin, N. J.; Singer, I. M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362, 1711, 425-461 (1978), MR 80d:53023 · Zbl 0389.53011
[6] Aubin, T., Nonlinear analysis on manifolds. Monge-Ampère Equations (1982), Springer: Springer New York, MR 85j:58002 · Zbl 0512.53044
[7] Bando, S., Bubbling out of Einstein manifolds, Tohoku Math. J. (2), 42, 2, 205-216 (1990), MR 92a:53065a · Zbl 0719.53025
[8] Bando, S., Correction and additionBubbling out of Einstein manifolds, Tohoku Math. J. (2), 42, 4, 587-588 (1990), MR 92a:53065b · Zbl 0762.53030
[9] Bando, S.; Kasue, A.; Nakajima, H., On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math., 97, 2, 313-349 (1989), MR 90c:53098 · Zbl 0682.53045
[10] Baston, R. J.; Mason, L. J., Conformal gravity, the Einstein equations and spaces of complex null geodesics, Classical Quantum Gravity, 4, 4, 815-826 (1987), MR 88h:83010 · Zbl 0647.53057
[11] Besse, A. L., Einstein Manifolds (1987), Springer: Springer Berlin · Zbl 0613.53001
[12] Borzellino, J. E., Orbifolds of maximal diameter, Indiana Univ. Math. J., 42, 1, 37-53 (1993), MR 94d:53053 · Zbl 0801.53031
[13] Bourguignon, J.-P., Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein, Invent. Math., 63, 2, 263-286 (1981), MR 82g:53051 · Zbl 0456.53033
[14] Branson, T., Kato constants in Riemannian geometry, Math. Res. Lett., 7, 2-3, 245-261 (2000), MR 2001i:58066 · Zbl 1039.53033
[15] Burago, D.; Burago, Y.; Ivanov, S., A course in metric geometry, (Graduate Studies in Mathematics, vol. 33 (2001), American Mathematical Society: American Mathematical Society Providence, RI), MR 2002e:53053 · Zbl 1040.53088
[16] Calabi, E., Extremal Kähler metrics, (Seminar on Differential Geometry (1982), Princeton University Press: Princeton University Press Princeton, NJ), 259-290, MR 83i:53088
[17] Calabi, E., Extremal Kähler Metrics, II, (Differential Geometry and Complex Analysis (1985), Springer: Springer Berlin), 95-114, MR 86h:53067 · Zbl 0574.58006
[18] Calderbank, D. M.J.; Gauduchon, P.; Herzlich, M., Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal., 173, 1, 214-255 (2000), MR 2001f:58046 · Zbl 0960.58010
[19] Carron, G., \(L^2\)-cohomologie et inégalités de Sobolev, Math. Ann., 314, 4, 613-639 (1999), MR 2000f:53045 · Zbl 0933.35054
[20] Cheeger, J.; Colding, T. H., On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom., 46, 3, 406-480 (1997), MR 98k:53044 · Zbl 0902.53034
[21] Cheeger, J.; Colding, T. H., On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom., 54, 1, 13-35 (2000), MR 1 815 410 · Zbl 1027.53042
[22] Cheeger, J.; Colding, T. H., On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom., 54, 1, 37-74 (2000), MR 1 815 411 · Zbl 1027.53043
[23] Cheeger, J.; Colding, T. H.; Tian, G., On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal., 12, 5, 873-914 (2002), MR 2003m:53053 · Zbl 1030.53046
[24] Cheeger, J.; Gromov, M.; Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17, 1, 15-53 (1982), MR 84b:58109 · Zbl 0493.53035
[25] Derdziński, A., Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math., 49, 3, 405-433 (1983), MR 84h:53060 · Zbl 0527.53030
[26] A. Derdziński, Riemannian manifolds with harmonic curvature, Global Differential Geometry and Global Analysis 1984 (Berlin, 1984), Springer, Berlin, 1985, pp. 74-85. MR 87d:53088.; A. Derdziński, Riemannian manifolds with harmonic curvature, Global Differential Geometry and Global Analysis 1984 (Berlin, 1984), Springer, Berlin, 1985, pp. 74-85. MR 87d:53088. · Zbl 0574.53032
[27] DeTurck, D. M.; Kazdan, J. L., Some regularity theorems in Riemannian geometry, Ann. Sci. Ecole Norm. Sup. (4), 14, 3, 249-260 (1981), MR 83f:53018 · Zbl 0486.53014
[28] Donaldson, S.; Friedman, R., Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity, 2, 2, 197-239 (1989), MR 90e:32027 · Zbl 0671.53029
[29] S.K. Donaldson, P.B. Kronheimer, The Geometry of Four-Manifolds, The Clarendon Press, Oxford University Press, New York, 1990, Oxford Science Publications. MR 92a:57036.; S.K. Donaldson, P.B. Kronheimer, The Geometry of Four-Manifolds, The Clarendon Press, Oxford University Press, New York, 1990, Oxford Science Publications. MR 92a:57036. · Zbl 0820.57002
[30] Floer, A., Self-dual conformal structures on \(l CP^2\), J. Differential Geom., 33, 2, 551-573 (1991), MR 92e:53049 · Zbl 0736.53046
[31] Freed, D. S.; Uhlenbeck, K. K., Instantons and Four-Manifolds (1991), Springer: Springer New York, MR 91i:57019 · Zbl 0559.57001
[32] Hebey, E., Sobolev spaces on Riemannian manifolds, (Lecture Notes in Mathematics, vol. 1635 (1996), Springer: Springer Berlin), MR 98k:46049 · Zbl 0866.58068
[33] Itoh, J.-I.; Tanaka, M., The dimension of a cut locus on a smooth Riemannian manifold, Tohoku Math. J. (2), 50, 4, 571-575 (1998), MR 99k:53068 · Zbl 0939.53029
[34] King, A. D.; Kotschick, D., The deformation theory of anti-self-dual conformal structures, Math. Ann., 294, 4, 591-609 (1992), MR 93j:58021 · Zbl 0765.58005
[35] LeBrun, C., Explicit self-dual metrics on \(CP_2 \# \cdots \# CP_2\), J. Differential Geom., 34, 1, 223-253 (1991), MR 92g:53040 · Zbl 0725.53067
[36] LeBrun, C., Thickenings and conformal gravity, Comm. Math. Phys., 139, 1, 1-43 (1991), MR 92h:83052 · Zbl 0733.53056
[37] C. LeBrun, Anti-self-dual metrics and Kähler geometry, Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 498-507. MR 97h:53049.; C. LeBrun, Anti-self-dual metrics and Kähler geometry, Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 498-507. MR 97h:53049. · Zbl 0838.53039
[38] Lee, J. M.; Parker, T. H., The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17, 1, 37-91 (1987) · Zbl 0633.53062
[39] Li, P.; Tam, L. F., Harmonic functions and the structure of complete manifolds, J. Differential Geom., 35, 2, 359-383 (1992), MR 93b:53033 · Zbl 0768.53018
[40] Merkulov, S. A., The twistor connection and gauge invariance principle, Comm. Math. Phys., 93, 3, 325-331 (1984), (MR 86c:53015) · Zbl 0549.53068
[41] Nakajima, H., Hausdorff convergence of Einstein 4-manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 35, 2, 411-424 (1988), MR 90e:53063 · Zbl 0655.53037
[42] Poon, Y. S., Compact self-dual manifolds with positive scalar curvature, J. Differential Geom., 24, 1, 97-132 (1986), MR 88b:32022 · Zbl 0583.53054
[43] Satake, I., On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. USA, 42, 359-363 (1956), MR 18,144a · Zbl 0074.18103
[44] Satake, I., The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan, 9, 464-492 (1957), MR 20 #2022 · Zbl 0080.37403
[45] Schoen, R., Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20, 2, 479-495 (1984) · Zbl 0576.53028
[46] R.M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Longman Science and Technology, Harlow, 1991, pp. 311-320. MR 94e:53035.; R.M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Longman Science and Technology, Harlow, 1991, pp. 311-320. MR 94e:53035.
[47] Schoen, R.; Yau, S.-T., Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math., 92, 1, 47-71 (1988), MR 89c:58139 · Zbl 0658.53038
[48] Schoen, R.; Yau, S.-T., Lectures on Differential Geometry (1994), International Press: International Press Cambridge, MA, MR 97d:53001 · Zbl 0830.53001
[49] Taubes, C. H., The existence of anti-self-dual conformal structures, J. Differential Geom., 36, 1, 163-253 (1992), MR 93jo:53063 · Zbl 0822.53006
[50] W.P. Thurston, S. Levy (Ed.), Three-dimensional Geometry and Topology, vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. MR 97m:57016.; W.P. Thurston, S. Levy (Ed.), Three-dimensional Geometry and Topology, vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. MR 97m:57016. · Zbl 0873.57001
[51] Tian, G., On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., 101, 1, 101-172 (1990), MR 91d:32042 · Zbl 0716.32019
[52] G.Tian, J. Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math., to appear, preprint, arXiv:math.DG/0310302.; G.Tian, J. Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math., to appear, preprint, arXiv:math.DG/0310302.
[53] G. Tian, S.-T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of String Theory (San Diego, CA, 1986), Advanced Series Mathematical Physics, vol. 1, World Scientific Publishing, Singapore, 1987, pp. 574-628. MR 915 840.; G. Tian, S.-T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of String Theory (San Diego, CA, 1986), Advanced Series Mathematical Physics, vol. 1, World Scientific Publishing, Singapore, 1987, pp. 574-628. MR 915 840.
[54] Uhlenbeck, K. K., Removable singularities in Yang-Mills fields, Comm. Math. Phys., 83, 1, 11-29 (1982), MR 83e:53034 · Zbl 0491.58032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.