An interesting (at least to me) research theme in the geometry of manifolds is the question about the existence of positive scalar curvature metrics on closed manifolds. Since I also like to do coarse geometry, I therefore also consider the corresponding question on non-compact manifolds. But what is the ‘corresponding’ question on non-compact manifolds?
Currently, there are two different variants of this question considered: One either asks about positive scalar curvature metrics or one asks about uniformly positive scalar curvature metrics (i.e., where the scalar curvature is everywhere bounded from below by a fixed positive number). On spin manifolds, the latter is considerably easier to handle than the former due to coarse index theory: If we have a complete metric of uniformly positive scalar curvature on a spin manifold \(M\), then the coarse index class \(\operatorname{Ind}(D) \in K_*(C^* M)\) of the spin Dirac operator vanishes. Here \(K_*(C^* M)\) denotes the K-theory of the Roe C*-algebra of \(M\). But such a vanishing result is not available if we only assume positivity of the scalar curvature, i.e., without a uniform lower bound for it, making it much harder to rule out the existence of such metrics.
Recently I stumbled about a paper by Nick Wright who actually considers a third variant (one which I myself did not consider up to now): He asks about properly positive scalar curvature metrics on a non-compact manifold \(M\), i.e., Riemannian metrics whose scalar curvature function is a proper function \(M \to [-t,\infty)\) for some number \(t\). This means that the scalar curvature must go to infinity as further we go ‘out’ in \(M\). (Note that the minus sign is deliberately written to indicate that on a compact subset the scalar curvature could be negative.)
His main result is that the question about properly positive scalar curvature (on spin manifolds) can be actually also attacked by coarse index theory: If we have a complete metric of properly positive scalar curvature on a spin manifold M, then the \(C_o\)-coarse index class \(\operatorname{Ind}_{C_0}(D) \in K_*(C^* M_{C_0})\) of the spin Dirac operator vanishes. Here \(K_*(C^* M_{C_0})\) denotes the K-theory of the Roe C*-algebra of \(M_{C_0}\), where the latter is the space \(M\) equipped with a non-standard coarse structure (called the \(C_0\)-coarse structure) in order to form the Roe C*-algebra.
Now I am wondering: Why is there a whole industry of mathematicians studying positive and uniformly positive scalar curvature metrics, but there is none studying properly positive scalar curvature? To me the latter seems as worth investigating as the other two.
edit: Christopher Wulff made me actually aware of a reason to study upsc-metrics on non-compact manifolds instead of the other two kinds. If one is interested on psc-metrics on closed manifolds, then one can often get good results by passing to the universal cover and investigating the pulled-back metric, which is then upsc. Hence if one can rule out the upsc-metric, one has also ruled out the existence of a psc-metric on the closed manifold. Of course, if one can even rule out a psc-metric on the universal cover, then one also gets the same conclusion on the closed manifold. But as I said, ruling out psc-metrics is considerably harder than upsc-metrics.