In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C 0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C 0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C 0 initial data.Date: September 12, 2019. Theorem 1.5 implies:Corollary 1.6. If (M, g) is a closed Riemannian manifold with C 0 metric g, and if there exists β ∈ (0, 1/2) such that, at every point in M , g has scalar curvature bounded below by κ in the β-weak sense, then there exists a sequence of C 2 metrics on M with scalar curvature bounded below by κ that converges uniformly to g.Theorem 1.5 also implies:Theorem 1.7. Let g be a C 0 metric on a closed manifold M which admits a uniform approximation by C 0 metrics g i such that, for some β ∈ (0, 1/2), g i has scalar curvature bounded below by κ i in the β-weak sense everywhere on M , where κ i is some sequence of numbers such that κ i − −− → i→∞ κ for some number κ. Then g has scalar curvature bounded below by κ in the β-weak sense. In particular, any regularizing Ricci flow (g(t)) t∈(0,T ] for g satisfies R(g(t)) ≥ κ for all t ∈ (0, T ], so g admits a uniform approximation by smooth metrics with scalar curvature bounded below by κ.As a corollary of Theorem 1.7, we may answer the following question, posed by Gromov in [8]:Question 1 ([8, Page 1119]). Let g be a continuous Riemannian metric on a closed manifold M which admits a C 0 -approximation by smooth Riemannian metrics g i with R(g i ) ≥ −ε i − −− → i→∞ 0. Does M admit a smooth metric of nonnegative scalar curvature?By setting κ i = −ε i in Theorem 1.7, we obtain the following response:Corollary 1.8. If (M, g) is as in Question 1, then any regularizing Ricci flow (g t ) t∈(0,T ] for g satisfies R(g t ) ≥ 0 for all t ∈ (0, T ]. In particular, M admits a smooth metric of nonnegative scalar curvature, and moreover, g admits a uniform approximation by smooth metrics with nonnegative scalar curvature.We use similar methods to show a torus rigidity result, motivated by the scalar torus rigidity theorem, which was first proven by Schoen and Yau [14] for dimensions ≤ 7, and later proven by Gromov and Lawson [7] for all dimensions, and which says that any Riemannian manifold with nonnegative scalar curvature that is diffeomorphic to the torus must be isometric to the flat torus. We show: Corollary 1.9. Suppose g is a C 0 metric on the torus T, and that there is some β ∈ (0, 1/2) such that g has nonnegative scalar curvature in the β-weak sense everywhere. Then (T, g) is isometric as a metric space to the standard flat metric on T.Remark 1.10. Corollary 1.9 is in fact the optimal result, i.e. it is not possible to show that there is a Riemannian isometry between g and the standard flat metric. In the case where g 1 and g 2 are smooth metrics, a metric space...

We survey some recent work using Ricci flow to create a class of local definitions of weak lower scalar curvature bounds that is well defined for C⁰ metrics. We discuss several properties of these definitions and explain some applications of this approach to questions regarding uniform convergence of metrics with scalar curvature bounded below. Finally, we consider the relationship between this approach and some other generalized notions of lower scalar curvature bounds.

No abstract