## any compactness theorem for manifolds which has Ricci lower bound?

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If manifolds have sectional curvature lower bound, then those manifolds has subsequence convergent to Alexandrov space. Is there similar results for manifolds with Ricci lower bound?

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Have you tried the Wikipedia page for Ricci curvature? en.wikipedia.org/wiki/Ricci_curvature – Ryan Budney Nov 27 2011 at 9:22

## 2 Answers

This is getting too long for a comment so I'm posting this as an answer...

@guoyi xu No you can not claim in any sense that $|\nabla_y dist_x|> 1- \delta$ for any `$y \in B_{b(x)}(x)\backslash \{x\}$` for limits of spaces with just lower Ricci bounds. That is indeed different from Alexandrov spaces where you do have this property. That makes studying limits of manifolds with lower Ricci bounds quite a bit harder than Alexandrov spaces. Let me give a rough reason why you can not possibly hope for such a result. In Alexandrov spaces the above property is a fairly easy consequence of Toponogov comparison. Toponogov comparison is false with just lower Ricci bounds. There is an $L^2$ version of Toponogov for long thin triangles due to Colding (and later Colding and Cheeger) but that is only $L^2$ which means it's just a measured statement which can not give you full control on topology (except in some near extremal cases) as there is always a small set of points where you can say nothing about the gradient of the distance function but which might happen to contain a lot of topological information. One may hope that maybe Cheeger--Colding results are not optimal and maybe the result you want could be proved using some undiscovered tools but that is not so. If you could prove such a statement for lower Ricci then you'd be able to prove the Gromov betti number estimate for lower Ricci curvature bound instead of lower sectional curvature bound. However, that is known to be false. There are many counterexamples in collapsing situation ( examples by Sha- Yang etc) and it's not even possible in the noncollapsing case. I think the first example in the noncollapsing case is by Perelman

Lastly, ( I should have probably started with that) there are some other pathalogical examples known which clearly violate the statement you are after. For example, Meguy constructed examples of limit spaces where some tangent cones are not polar meaning that not every geodesic starting at the origin can be extended to a ray.

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@Vitali Kapovitch, I should be more specific here. I am focusing on 3-dimensional case, by Hamilton, Yau-schoen and recent Liu Gang's result, 3-dim manifolds with nonnegative Ricci curvature has finite topology type, so I guess my statement about limit space of 3-dim noncompact manifolds with nonnegative Ricci curvature could be true although I prefer more direct argument not using classification results. What do you think? – guoyi xu Nov 28 2011 at 5:49

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Dimension 3 is certainly very special in a number of ways but I really don't expect the statement you are after to be true even there. Liu Gang's beautiful result (and the original paper by Schoen and Yau) is much more subtle and uses very different ideas which I don't think can be easily circumvented. – Vitali Kapovitch Nov 28 2011 at 13:46

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A well-known theorem of Gromov says that if $(M_i,g_i)$ is a sequence of manifolds with $Ric\geq -kg$ and fixed dimension, then it is a converging subsequence in the pointed Gromov-Hausdorff topology. References for this fact are Gromov's "Metric Structures for Riemannian and non-Riemannian spaces" and Burago-Burago-Ivanov's "A Course in Metric Geometry".

A lot of investigation has been made on the possible limiting length space, see "On the structure of spaces with Ricci curvature bounded below." by Cheeger and Colding, and subsequent papers.

On the other hand, one can have smoother convergence with stronger geometric requirements. For instance, one has $C^\alpha$ compactness when the injectivity radius is bounded from below, see "$C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below", by Anderson and Cheeger.

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In fact, precompactness of the class of $n$-manifolds with $Ric\ge k$ curvature bounds was proved by Gromov directly (sectional curvature is just a special case). It's an easy consequence of Bishop-Gromov volume comparison which implies a universal bound in terms of $\epsilon, k,n, R$ on the number of $\epsilon$-separated points in any $R$ ball. Such a bound for a class of proper inner metric spaces implies precompactness in pointed Gromov-Hausdorff topology. – Vitali Kapovitch Nov 27 2011 at 14:35
In fact, I am interested in whether the limit space's distance function has some C^1 property or critical point theory or not like Alexandrov space has. – guoyi xu Nov 27 2011 at 19:18

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@guoyi xu No. there is no $C^1$ structure on limit spaces with lower Ricci curvature bounds - they can be extremely singular (there isn't one even for Alexandrov spaces btw) and there is no critical point theory either. Critical point theory for Alexandrov spaces is possible basically because distance functions on Alexandrov spaces are semiconcave. Distance functions on manifolds with lower Ricci curvature bounds are only semi- super harmonic. This is a much weaker condition which doesn't allow for much control of topology except on the level of $\pi_1$ and in near extremal cases. – Vitali Kapovitch Nov 27 2011 at 19:43
@Vitali Kapovitch, so for such limit space, can we claim that there exists a positive function b(x) on this space, such that |\nabla_y dist_x|> 1- \delta for any y \in B_{b(x)}(x)\{x}? In some sense, could we define spire on such limit space like in Alexandrov spaces? – guoyi xu Nov 27 2011 at 23:04
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