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The collapsing geometry of almost Ricci-flat 4-manifolds. (English) Zbl 1453.53046

Let \(M\) be a compact 4-manifold that admits a sequence of Riemannian metrics \(\{g_i\}\) with \(\lim \left(\|\text{Ric}(M,g_i)\|_\infty\cdot\text{diam}(M,g_i)^2\right)=0\). That is, \(M\) is almost Ricci-flat. If one fixes an upper diameter bound for the sequence \((M,g_i)\), then one can study the non-collapsing case \(\text{vol}((M,g_i))\ge v>0\), and the collapsed case where \(\text{vol}((M,g_i))\to 0\). Gromov-Hausdorff limits in the non-collapsing case were studied by M. T. Anderson [Invent. Math. 102, No. 2, 429–445 (1990; Zbl 0711.53038)], S. Bando et al. [Invent. Math. 97, No. 2, 313–349 (1989; Zbl 0682.53045)] and G. Tian [Invent. Math. 101, No. 1, 101–172 (1990; Zbl 0716.32019)]. The collapsing case was studied by J. Cheeger and G. Tian [J. Am. Math. Soc. 19, No. 2, 487–525 (2006; Zbl 1092.53034)]. Suppose now that the manifolds are allowed to vary and let \((M_i,g_i)\) be a sequence of Riemannian 4-manifolds so that for some \(C\in\mathbb{N}\) and \(D<\infty\) the following four conditions hold: (1) \(\chi(M_i)\le C\), (2) \(\text{diam}(M,g_i)\le D\), (3) \(\lim \left\|\text{Ric}(M,g_i)\right\|_\infty=0\), and (4) \(\lim \text{vol}((M,g_i))=0\). The Gromov-Hausdorff limit of a subsequence will converge to a compact metric space \(X\) with Hausdorff dimension less than four. The work of Cheeger and Tian showed that under these conditions the subsequence will experience regions where there is a small curvature blowup and a “regular” region where there are a priori curvature bounds. In particular, the regular regions converge to a subset \(X_{\text{reg}}\subset X\) and the metric completion of a connected component \(B\) of \(X_{\text{reg}}\) consists of adding a finite number of points. The main result of this paper characterizes the geometry of the subsets \(B\) according to their dimension.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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