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Suppose $\left( {{M_i},{q_i}} \right)\mathop \to \limits^{G - H} \left( {X,{q_\infty }} \right)$, ${\rm Ric}_{M_i} \ge - \left( {n - 1} \right)$, and ${q_\infty }$ is regular, i.e. all the tangent cones are equal to ${R^k}$ where $k$ depends on $p$.

Then can we find a sequence ${\lambda _i} \to \infty $,such that $\left( {{\lambda _i}{M_i},{q_i}} \right)\mathop \to \limits^{G - H} \left( {{R^k},0} \right)$? And is there an example where some point of the limit place has non-unique tangent cones?

This question may repeat a question I have asked. Please forgive me.

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Unless I've misunderstood the question, isn't the answer to the first question obviously yes? We have $(M_i, q_i) \to (X, q_\infty)$ and $(\lambda_i X, q_\infty)\to(R^k,0)$ by assumption. –  Dan Lee Jul 6 '13 at 21:03

1 Answer 1

Tangent cones to noncollapsed limit spaces (with lower Ricci bounds) need not be unique. This was first observed by Perelman (more precisely, an analogous statement for asymptotic cones) and then Cheeger and Colding (JDG). Although there might be different tangent cones, all of them must be metric cones over a compact metric space with diameter $\leq\pi$ and Hausdorff dimension $n-1$. Recently, Colding and Naber (GAFA) gave a detailed description of what spaces can arise as tangent cones to a noncollapsed limit with lower Ricci bounds. All answers to your questions above can be found in this last reference.

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Hi Renato, I think the answer is potentially different if the tangent cone is Euclidean space (which I think is what the OP meant by "$q_\infty$ is regular") In particular, for the analogous question for minimal surfaces, if the tangent cone is a (multiplicity one) plane, then Allard's theorem implies local regularity, so the GMT tangent cones must all agree with the "classical" tangent cone and thus be unique. I'm not sure one can say here, especially if $k < n$, however. –  Otis Chodosh Jun 28 '13 at 20:32
    
Otis, the OP defines "$q_\infty$ is regular" to mean that R^k is the unique tangent cone at $q_\infty$, so I think that OP's second question is supposed to be unrelated to that bit. (But it's not so clearly written.) –  Dan Lee Jul 6 '13 at 21:08

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