Recently, Guo and Yu pushed the following result to the arXiv (math.KT/2203.15003):
For any \(R > 0\) and positive integer \(m\), there exists a constant \(k(R,m)\) such that the following holds. If \((M,g)\) is a Riemannian spin manifold that admits a uniformly bounded, good open cover with Lebesgue number \(R\) and \(R\)-multiplicity \(m\), then \[\inf_{x \in M} \kappa_g(x) \le k(R,m)\,,\] where \(\kappa_g\) is the scalar curvature of \(g\). Here an open cover \(\mathcal{U} = \{U_i\}_{i \in I}\) is called good if any finite intersection of its members is contractible.
I immediately had the feeling (which was strengthened by the title of the preprint) that this is a quantitative version of the following result: If \((M,g)\) is a uniformly contractible Riemannian spin manifold of finite asymptotic dimension, then \(g\) does not have uniformly positive scalar curvature. (The occurring notions will be explained further below.)
In this blog post I want to explain the relation between these two results. First of all, to use the theorem of Guo-Yu to rule out \(g\) having uniformly positive scalar curvature, we have to assume that \((M,g)\) admits for a fixed \(m\) and any \(R > 0\) a covering as stated in the theorem. This is because the constant \(k(R,m)\) depends quadratically on \(m\) and inverse quadratically on \(R\), i.e., for a fixed \(m\) we have \[k(R,m) \xrightarrow{R \to \infty} 0\,.\]
Now assume that we have a Riemannian manifold \((M,g)\) that admits for a fixed \(m\) and any \(R > 0\) a uniformly bounded (open) cover with Lebesgue number \(R\) and \(R\)-multiplicity \(m\). Note that we do not assume the cover to be good here. Then this means that \((M,g)\) has finite asymptotic dimension (bounded from above by \(m\)); in fact, this is just an equivalent reformulation of it.
Recall that a Riemannian manifold \((M,g)\) is uniformly contractible, if for every \(x \in M\) the ball \(B_r(x)\) of radius \(r\) is contractible inside the larger ball \(B_s(x)\) of radius \(s\). Now assume that \((M,g)\) admits for a fixed \(m\) and any \(R > 0\) a covering as stated in the theorem of Guo-Yu. Then any ball \(B_r(x)\) is contained in a member \(U_{x}\) of the uniformly bounded, good open cover \(\mathcal{U}\) with Lebesgue number \(r\). Since this cover is good, \(U_{x}\) is contractible and hence \(B_r(x)\) is contractible inside of it. If \(s\) now denotes the upper bound on the diameters of members of \(\mathcal{U}\), then this means that \(B_r(x)\) is contractible inside \(B_s(x)\). This means that \((M,g)\) is uniformly contractible, and we have already seen above that it also has finite asymptotic dimension.
The other direction, i.e., if \((M,g)\) is uniformly contractible and has finite asymptotic dimension, that it then must admit for a fixed \(m\) and any \(R > 0\) a covering as stated in the theorem of Guo-Yu, should be also true, but after 10 seconds of thinking I couldn’t come up with an argument (maybe I should just think a bit more about it …). If you see one, feel free to post it in the comments.