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: tian@math.mit.edu (G. Tian), jeffv@math.mit.edu (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

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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 -