# Ricci curvature under rough convergence

From the work of Lott--Villani and Sturm, I know that the following fact holds:

(*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci curvature that converges in the measured Gromov--Hausdorff sense to $(M_\infty,g_\infty,dvol_{g_\infty})$, which is a smooth Riemannian manifold. Then $(M_\infty,g_\infty)$ also has non-negative Ricci curvature.

The proof I have in mind goes like: $(M_k,g_k,dvol_{g_k})$ has non-negative Ricci curvature in the weak optimal transport sense. This is preserved under measured Gromov--Hausdorff convergence. Hence $(M_\infty,g_\infty,dvol_{g_\infty})$ has non-negative Ricci curvature in the optimal transport sense. Because it is smooth, this then implies that it has non-negative Ricci curvature in the usual sense.

I think, however, that this fact was known long before the optimal transport interpretation of Ricci lower bounds (there were lots of works on GH convergence and lower Ricci bounds, e.g. Cheeger and Colding had a sequence of papers studying the regularity of such limits (the first one is here). However, I could not find a reference for this. So, my question is:

Can (*) be proven without the optimal transport theory?

EDIT: It seems like there was some ambiguity in my question, as I did not specify if the sequence was allowed to collapse to a lower dimension. I had the case of non-collapsing in mind, but am very interested in the general case as well. So, to summarize, I believe that the non-collapsing case (i.e. $M_k$ and $M_\infty$ have the same dimension), there was a well known proof, while in the collapsed case, optimal transport provided the first proof.

• I had the idea that this was not known before, and in fact the most concrete consequence of Lott-Villani-Sturm's work (as we do not have plenty of non-manifolds satisfying CD(k,N) and the like). But I may very well be wrong; did you check whether Lott Villani and Sturm allude something about this application of their work? Mar 22, 2014 at 18:45
• Hi Benoit! I guess its possible you are right: e.g. Lott--Villani do include this as Corollary 0.14 in their paper. Mar 22, 2014 at 19:24
• @BenoîtKloeckner, if you want to add this as an answer, I'll wait a few days to see if anyone has any other ideas and accept yours if not! Mar 22, 2014 at 22:12

Let me try to prove it.

Let $M$ be an $m$-dimensional Riemannian manifold. Set $$f_p(z)=\frac{|p-z|_M^2}2,$$ where $|p-z|_M$ denotes the distance from $p$ to $z$ in $M$.

Note that $M$ has nonnegative Ricci curvature if and only if $$(\Delta f_p)(x)\le m$$ for any $p$ and $x\in M$. The later means that for the integral $$\int\limits_{B_r(x)}f_p(z)\cdot d_z\mathrm{vol}$$ the comparison inequality holds, i.e., if $\tilde p,\tilde x\in\mathbb{E}^m$ and $f_{\tilde p}(\tilde x)=f_{p}(x)$ then $$\int\limits_{B_r(x)\subset M}f_p(z)\cdot d_z\mathrm{vol} \le \int\limits_{B_r(\tilde x)\subset \mathbb{E}^m}f_{\tilde p}(\tilde z)\cdot d_{\tilde z}\mathrm{vol}.$$

The last inequality survives in measured Gromov--Hausdorff limit.

• P.S. also, it should be known that if $m$-dimensional manifolds with lower bound for Ricci curvature GH-converge to an $m$-dimensional manifold then it converges in measured Gromov--Hausdorff sense. It is Colding's result if I remember right, likely he also proved the statement you asked. Mar 23, 2014 at 19:31
• Thanks! This is excellent. I believe jstor.org/stable/2951841 is the paper you mean. Mar 23, 2014 at 22:21
• @AntonPetrunin, I spoke to John Lott, who remarked that there is an ambiguity in what the OP is after. The easy case is where the manifolds are of the same dimension. But his proof can handle the case where the limiting manifold has a different dimension (and for which he is unaware of an alternative proof). Mar 24, 2014 at 7:05
• @ChrisGerig, thanks for your comment! I was thinking of "easy case" when I asked the question, but am interested in general! Mar 24, 2014 at 14:08
• @ChrisGerig, I do not see ambiguity in the question. In fact it takes some effort to formulate the collapsing version of the statement without optimal transport (try to do this). Mar 24, 2014 at 17:51

(As asked, I make my comment into an answer, completing it with the OP's comment)

As far as I know, this result was not known before Lott-Villani and Sturm's works; it is included as Corollary 0.14 in Lott-Villani's article.

As we do not know a lot of spaces satisfying the Ricci lower bounds defined by this theory, apart from manifolds and Alexandrov spaces, this is currently the most concrete application of these works.

Added: for clarity, I stress here the comment of Chris Gerig to Anton Petrunin's answer: Lott-Villani and Sturm's prove that the result holds even if the limiting manifold is of different dimension than the converging manifolds.

• I added a remark in the question. Thanks for your answer! I apologize for the ambiguity in my question. Mar 24, 2014 at 14:18