Pointwise lower scalar curvature bounds for $$C^0$$ metrics via regularizing Ricci flow

Abstract

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...

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

“…Recently, Burkhardt-Guim [6] proposed a different possible notion of "R > κ" using Ricci flow. See [6,Definition 1.2]. These definitions all share some good properties as a weak notion.…”

Defining Pointwise Lower Scalar Curvature Bounds for C⁰ Metrics with Regularization by Ricci Flow

“…The existence of a solution to (2.4) is proven in a similar fashion to [7]: in [7] Koch and Lamm construct suitable Banach spaces so that a solution to (2.4) arises from an application of the Banach fixed point theorem. In [3] one performs a similar construction, working on a Ricci flow background rather than a stationary background. Once the existence of a solution to the integral equation (2.4) has been established, smoothness of the solution and bounds on the derivatives may be proven by iterative application of parabolic interior estimates.…”

“…Remark 3.6. Definition 3.4 may also be formulated in terms of Ricci-DeTurck flow (see [3,Lemma 6.4]). The equivalent statement using Ricci-DeTurck flow is that there is a Ricci-DeTurck flow g t starting from g 0 in the sense of Proposition 2.1 andḡ 0 a stationary metric on M that is uniformly bilipschitz to (g t ) t∈(0,T ] such that…”

“…In light of Bamler's approach to Theorem 1.1, it is natural to ask whether it is possible to use Ricci flow to formulate a generalized definition of lower scalar curvature bounds for C 0 metrics. In [3] the author proposed a class of such definitions, and proved some related results. Generally speaking, the lower scalar curvature bounds in [3] are determined by using the Ricci flow to "regularize" the singular metric and then observing the scalar curvature of the flow at small positive times.…”

“…In [3] the author proposed a class of such definitions, and proved some related results. Generally speaking, the lower scalar curvature bounds in [3] are determined by using the Ricci flow to "regularize" the singular metric and then observing the scalar curvature of the flow at small positive times. The purpose of this survey is to describe more precisely how to formulate such a definition, to discuss several natural properties of this definition, and to explain how to prove some stability and rigidity results concerning the uniform convergence of metrics satisfying certain lower scalar curvature bounds, such as the following: Theorem 1.2.…”