TY - JOUR
T1 - Moduli spaces of critical Riemannian metrics in dimension four
AU - Tian, Gang
AU - Viaclovsky, Jeff
N1 - Funding Information:
∗ Corresponding author. Fax: +1 617 253 4358. E-mail addresses: [email protected] (G. Tian), [email protected] (J. Viaclovsky). 1Current address: Department of Mathematics, Princeton University, Princeton, NJ 08544, 2The research of the first author was partially supported by NSF Grant DMS-0302744. 3The research of the second author was partially supported by NSF Grant DMS-0202477.
PY - 2005/10/1
Y1 - 2005/10/1
N2 - 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 (J. Amer. Math. Soc. 2(3) (1989) 455, Invent. Math. 97 (2) (1989) 313, Invent. Math. 101(1) (1990) 101), but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound.
AB - 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 (J. Amer. Math. Soc. 2(3) (1989) 455, Invent. Math. 97 (2) (1989) 313, Invent. Math. 101(1) (1990) 101), but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound.
KW - Anti-self-dual metrics
KW - Orbifolds
UR - http://www.scopus.com/inward/record.url?scp=24044470994&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=24044470994&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2004.09.004
DO - 10.1016/j.aim.2004.09.004
M3 - Article
AN - SCOPUS:24044470994
SN - 0001-8708
VL - 196
SP - 346
EP - 372
JO - Advances in Mathematics
JF - Advances in Mathematics
IS - 2
ER -