Moduli spaces of critical Riemannian metrics in dimension four

Advances in Mathematics

Volumn 196, Issue 2, 2005, Pages 346-372

  Tian, Gang ^a,b   Viaclovsky, Jeff ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b PRINCETON UNIVERSITY   (United States)

Author keywords

Anti self dual metrics; Orbifolds

Indexed keywords

EID: 24044470994     PISSN: 00018708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.aim.2004.09.004     Document Type: Article

References (54)

1. K. Akutagawa Yamabe metrics of positive scalar curvature and conformally flat manifolds Differential Geom. Appl. 4 3 1994 239-258 MR 95i:58046
2. M.T. Anderson Orbifold compactness for spaces of Riemannian metrics and applications preprint arXiv:mAth.DG/0312111
3. M.T. Anderson Ricci curvature bounds and Einstein metrics on compact manifolds J. Amer. Math. Soc. 2 3 1989 455-490 MR 90g:53052
4. V. Apostolov D.M.J. Calderbank P. Gauduchon The geometry of weakly self-dual Kähler surfaces Compositio Math. 135 3 2003 279-322 MR 1 956 815
5. M.F. Atiyah N.J. Hitchin I.M. Singer Self-duality in four-dimensional Riemannian geometry Proc. Roy. Soc. London Ser. A 362 1711 1978 425-461 MR 80d:53023
6. T. Aubin Nonlinear analysis on manifolds. Monge-Ampère Equations 1982 Springer New York MR 85j:58002
7. S. Bando Bubbling out of Einstein manifolds Tohoku Math. J. (2) 42 2 1990 205-216 MR 92a:53065a
8. S. Bando Correction and addition Bubbling out of Einstein manifolds Tohoku Math. J. (2) 42 4 1990 587-588 MR 92a:53065b
9. 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 2 1989 313-349 MR 90c:53098
10. R.J. Baston L.J. Mason Conformal gravity, the Einstein equations and spaces of complex null geodesics Classical Quantum Gravity 4 4 1987 815-826 MR 88h:83010
11. A.L. Besse Einstein Manifolds 1987 Springer Berlin
12. J.E. Borzellino Orbifolds of maximal diameter Indiana Univ. Math. J. 42 1 1993 37-53 MR 94d:53053
13. J.-P. Bourguignon Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein Invent. Math. 63 2 1981 263-286 MR 82g:53051
14. T. Branson Kato constants in Riemannian geometry Math. Res. Lett. 7 2-3 2000 245-261 MR 2001i:58066
15. D. Burago Y. Burago S. Ivanov A course in metric geometry Graduate Studies in Mathematics 33 2001 American Mathematical Society Providence, RI MR 2002e:53053
16. E. Calabi Extremal Kähler metrics Seminar on Differential Geometry 1982 Princeton University Press Princeton, NJ 259-290 MR 83i:53088
17. E. Calabi Extremal Kähler Metrics, II Differential Geometry and Complex Analysis 1985 Springer Berlin 95-114 MR 86h:53067
18. D.M.J. Calderbank P. Gauduchon M. Herzlich Refined Kato inequalities and conformal weights in Riemannian geometry J. Funct. Anal. 173 1 2000 214-255 MR 2001f:53046
19. 2-cohomologie et inégalités de Sobolev Math. Ann. 314 4 1999 613-639 MR 2000f:53045
20. J. Cheeger T.H. Colding On the structure of spaces with Ricci curvature bounded below, I J. Differential Geom. 46 3 1997 406-480 MR 98k:53044
21. J. Cheeger T.H. Colding On the structure of spaces with Ricci curvature bounded below, II J. Differential Geom. 54 1 2000 13-35 MR 1 815 410
22. J. Cheeger T.H. Colding On the structure of spaces with Ricci curvature bounded below, III J. Differential Geom. 54 1 2000 37-74 MR 1 815 411
23. J. Cheeger T.H. Colding G. Tian On the singularities of spaces with bounded Ricci curvature Geom. Funct. Anal. 12 5 2002 873-914 MR 2003m:53053
24. 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 1 1982 15-53 MR 84b:58109
25. A. Derdziński Self-dual Kähler manifolds and Einstein manifolds of dimension four Compositio Math. 49 3 1983 405-433 MR 84h:53060
26. A. Derdziński Riemannian manifolds with harmonic curvature, Global Differential Geometry and Global Analysis 1984 (Berlin 1984) Springer Berlin 1985 74-85 MR 87d:53088
27. D.M. DeTurck J.L. Kazdan Some regularity theorems in Riemannian geometry Ann. Sci. Ecole Norm. Sup. (4) 14 3 1981 249-260 MR 83f:53018
28. S. Donaldson R. Friedman Connected sums of self-dual manifolds and deformations of singular spaces Nonlinearity 2 2 1989 197-239 MR 90e:32027
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
30. 2 J. Differential Geom. 33 2 1991 551-573 MR 92e:53049
31. D.S. Freed K.K. Uhlenbeck Instantons and Four-Manifolds second ed. 1991 Springer New York MR 91i:57019
32. E. Hebey Sobolev spaces on Riemannian manifolds Lecture Notes in Mathematics vol. 1635 1996 Springer Berlin MR 98k:46049
33. J.-I. Itoh M. Tanaka The dimension of a cut locus on a smooth Riemannian manifold Tohoku Math. J. (2) 50 4 1998 571-575 MR 99k:53068
34. A.D. King D. Kotschick The deformation theory of anti-self-dual conformal structures Math. Ann. 294 4 1992 591-609 MR 93j:58021
35. 2 J. Differential Geom. 34 1 1991 223-253 MR 92g:53040
36. C. LeBrun Thickenings and conformal gravity Comm. Math. Phys. 139 1 1991 1-43 MR 92h:83052
37. C. LeBrun Anti-self-dual metrics and Kähler geometry Proceedings of the International Congress of Mathematicians 1 2 (Zürich 1994) Birkhäuser Basel 1995 498-507 MR 97h:53049
38. J.M. Lee T.H. Parker The Yamabe problem Bull. Amer. Math. Soc. (N.S.) 17 1 1987 37-91
39. P. Li L.F. Tam Harmonic functions and the structure of complete manifolds J. Differential Geom. 35 2 1992 359-383 MR 93b:53033
40. S.A. Merkulov The twistor connection and gauge invariance principle Comm. Math. Phys. 93 3 1984 325-331 (MR 86c:53015)
41. H. Nakajima Hausdorff convergence of Einstein 4-manifolds J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 2 1988 411-424 MR 90e:53063
42. Y.S. Poon Compact self-dual manifolds with positive scalar curvature J. Differential Geom. 24 1 1986 97-132 MR 88b:32022
43. I. Satake On a generalization of the notion of manifold Proc. Nat. Acad. Sci. USA 42 1956 359-363 MR 18,144a
44. I. Satake The Gauss-Bonnet theorem for V-manifolds J. Math. Soc. Japan 9 1957 464-492 MR 20 #2022
45. R. Schoen Conformal deformation of a Riemannian metric to constant scalar curvature J. Differential Geom. 20 2 1984 479-495
46. R.M. Schoen On the number of constant scalar curvature metrics in a conformal class Differential geometry, Longman Science and Technology Harlow 1991 311-320 MR 94e:53035
47. R. Schoen S.-T. Yau Conformally flat manifolds, Kleinian groups and scalar curvature Invent. Math. 92 1 1988 47-71 RM 89c:58139
48. R. Schoen S.-T. Yau Lectures on Differential Geometry 1994 International Press Cambridge, MA MR 97d:53001
49. C.H. Taubes The existence of anti-self-dual conformal structures J. Differential Geom. 36 1 1992 163-253 MR 93jo:53063
50. W.P. Thurston S. Levy (Ed.) Three-dimensional Geometry and Topology 1 Princeton Mathematical Series 35 Princeton University Press Princeton NJ 1997 MR 97m:57016
51. G. Tian On Calabi's conjecture for complex surfaces with positive first Chern class Invent. Math. 101 1 1990 101-172 MR 91d:32042
52. 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 574-628 MR 915 840
54. K.K. Uhlenbeck Removable singularities in Yang-Mills fields Comm. Math. Phys. 83 1 1982 11-29 MR 83e:53034