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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.

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