Bach-flat asymptotically locally Euclidean metrics

Abstract

We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results are known for Einstein metrics, but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.Comment: 54 pages; final versio

A Concentration‐Collapse Decomposition for L2 Flow Singularities

“…In [40] the authors show ǫ-regularity for critical metrics assuming a local Sobolev constant bound, and in [42] this is reduced to assuming a lower volume growth bound, as we have done here. A crucial issue in ( [40,41,42]) is that the critical equation does not automatically imply a Ricci curvature bound, and obtaining this bound and the attendant volume comparison bounds requires great care. In our parabolic case this issue is compounded due to the fact that the metric is changing in such a way that has no obvious C 0 control.…”

“…This is similar in some ways to the extra terms which arise in showing ǫ-regularity for extremal Kähler metrics as in [16]. Roughly speaking, the method in [40] is to couple (5.1) to the general equation satisfied by any Riemannian metric,…”

“…What differentiates Theorem 1.24 from the results of [7,40] is that we do not assume constant scalar curvature, which follows automatically in the settings considered in those papers. In some of our blowup arguments we obtain only a local limit, and thus one cannot ensure that the scalar curvature is constant.…”

“…Remark 1.11. The elliptic version of these estimates comes from work of Tian-Viaclovsky ( [40,41,42] cf. Chang [7]).…”

“…Moreover, due to the lack of a priori Ricci curvature bounds, in the course of the corresponding arguments in [40] a preliminary Ricci curvature bound must be obtained, which is essential in controlling volume growth using Bishop-Gromov volume comparison, and in turn obtaining the higher order estimates. In our setting this is not possible, and so instead we first obtain an L p bound on Ricci curvature which allows the application of volume growth estimates due to Petersen-Wei [32].…”