Commentarii Mathematici Helvetici

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Volume 83, Issue 4, 2008, pp. 889–911

DOI: 10.4171/CMH/147

Volume growth, curvature decay, and critical metrics

Gang Tian (1) and Jeff Viaclovsky (2)

(1) Department of Mathematics, Princeton University, Fine Hall, Washington Road, NJ 08544-1000, PRINCETON, UNITED STATES
(2) Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, WI 53706-1388, MADISON, UNITED STATES

We make some improvements to our previous results in [TV05a] and [TV05b]. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. As an application of these results, we can weaken the assumptions of several of our theorems in [TV05a] and [TV05b].

Keywords: Anti-self-dual metrics, ALE spaces, curvature decay, Îµ-regularity, orbifold compactness, volume growth