Volume growth, curvature decay, and critical metrics

Abstract

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].