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On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. (English) Zbl 0814.53034
Let $$M^ n$$ be a complete Riemannian manifold with metric $$g$$, such that $$\text{Ric}_ M \geq 0$$, $$\text{Vol} (B_ r(p)) \geq \Omega r^ n$$ for some constant $$\Omega > 0$$. Fix $$p \in M$$ and $$r_ j \to \infty$$. Then the sequence of pointed rescaled manifolds, $$(M,p, r_ j^{-2}g)$$, has a subsequence which converges in the pointed Gromov-Hausdorff topology to a length space $$M_ \infty$$. A basic question concerning $$M_ \infty$$ is whether or not it is unique, i.e. is $$M_ \infty$$ the same up to isometry for all $$\{r_ j\}$$ and all convergent subsequences. In general, it is not true even under the additional condition of quadratic curvature decay. In the present paper, for $$M$$ assumed to be Ricci flat, some sufficient condition for uniqueness of $$M_ \infty$$ is given. Moreover, the authors prove an analogous result for $$M$$ being a Kähler manifold.

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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