# A quantitative coarse obstruction to psc-metrics

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.