1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – Dan Lee
    Jul 6, 2013 at 21:03

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$ Jun 28, 2013 at 20:32
  • $\begingroup$ 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.) $\endgroup$
    – Dan Lee
    Jul 6, 2013 at 21:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.