any compactness theorem for manifolds which has Ricci lower bound? - MathOverflow

any compactness theorem for manifolds which has Ricci lower bound?

2011-11-27T09:15:14Z

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?

Answer by Thomas Richard for any compactness theorem for manifolds which has Ricci lower bound?

2011-11-27T11:20:48Z

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.

Answer by Vitali Kapovitch for any compactness theorem for manifolds which has Ricci lower bound?

2011-11-28T03:26:45Z

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.