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Regularity of Einstein manifolds and the codimension 4 conjecture. (English) Zbl 1335.53057
The authors consider the class of pointed Riemannian manifolds \((M^n,g,p)\) with bounded Ricci curvature \(|\mathrm{Ric}|\leq n-1\) which satisfies the noncollapsing assumption about the radius-one ball \(\mathrm{Vol}(B_1(p)) >v >0 \). They are interested in the properties of pointed Gromov-Hausdorff limits \[ (M^n_j,d_j,p_j) \to (X,d,p) \] of a sequence of such manifolds with respect to the Riemannian distance \(d_j\). There are many deep results. The most important is a proof of the following codimension-4 conjecture: Theorem. The singular set \(S\) of the limit space \(X\) has Hausdorff or Minkowski dimension \(\dim S \leq n-4\).
Combining this result with the ideas of quantitative stratification, the authors get an a priori \( L^q\) estimate on the full Riemannian curvature for all \( q < 2\). In the case of Einstein manifolds, they improve this to estimate the regularity scale. These results are applied for proving the following conjecture by Anderson about 4-dimensional Riemannian manifolds:
Theorem. There exists a constant \(C = C(v, D)\) such that if \( M^4\) satisfies \(|\mathrm{Ric}_{M^4}|\leq 3\), \(\mathrm{Vol}(B_1(p)) > v > 0\) and \(\operatorname{diam}(M^4) \leq D\), then \( M^4\) can have one of at most \( C\) diffeomorphism types.
The authors conjecture that this result holds in all dimensions.
It is shown also that noncollapse Riemannian 4-manifolds with bounded Ricci curvature admit an a priori \(L^2\) estimate on the Riemannian curvature tensor.

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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