This is a follow up to a question by Yiyue Zhang. Can scalar curvature and diameter control volume?
The original question asked whether scalar curvature bounds and small diameter bounds were enough to ensure that the volume of a manifold is less than that of a sphere. It turns out that one can create counter-examples to this by allowing negative Ricci curvature in some directions.
If one adds the assumption that the Ricci curvature is non-negative, I was wondering whether it is possible to get volume bounds that are stronger than the volume comparison theorem to flat space. More precisely, are there manifolds of dimension $n$ which have the following properties?
- The diameter of $M$ is bounded above by $\pi$. In other words, the diameter is smaller than that of a unit $n$-sphere.
- The scalar curvature of $M$ is bounded below by $n(n-1)$. In other words, the scalar curvature is greater than that of a unit $n$ sphere.
- The Ricci curvature of $M$ is non-negative.
- And finally, the volume of $M$ is greater than that of a unit $n$-sphere.
If such examples exist, what is the maximal volume of such a manifold? In essence, this is asking for manifolds with small diameter, large scalar curvature and non-negative Ricci but still relatively large volume.
A related question is what is the maximal volume among metrics of diameter $d$, scalar curvature at least $n(n-1)$ and non-negative Ricci? Here, we don't force the diameter to be the same as that of a unit sphere.
Some relevant information
After experimenting with the metric products of spheres and tori, I was unable to create such examples. Using Stirling's approximation and some other estimates, it may be possible to prove that such examples do not exist for metric products of spheres. However, if one can build in flat sections more carefully, then it may well be possible to cook up examples.
It is worth noting that when the scalar curvature is large, one can prove volume comparison theorems with less tight bounds on the Ricci curvature. For an example of this, see the answer to this question. Counterexample to volume comparison inequality assuming only scalar curvature bound?
When the Ricci is non-negative and the diameter is bounded, the volume is bounded above by a comparison to flat space. Using Gromov's compactness theorem, a sequence of such metrics with volumes tending towards the supremum has a convergent subsequence in Gromov-Hausdorff sense. However, it is not clear to me that the limiting space would have the necessary scalar curvature bounds.
When $d$ is large in the second question, one candidate for the largest volume is the metric product of a $2$-sphere with scalar curvature $n(n-1)$ and a flat torus of dimension $n-2$. However, I have no idea how to prove this. I also don't know of any other good examples that have large scalar curvature, bounded diameter, and fairly large volume.