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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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