Scalar curvature lower bound under integral convergence

Abstract

In this work, we consider sequences of C 2 metrics which converges to a C 2 metric in C 0 sense. We show that if the scalar curvature of the sequence is almost non-negative in the integral sense, then the limiting metric has scalar curvature lower bound in point-wise sense.

$C^{0},\, C^{1}$-limit theorems for total scalar curvatures

“…On the other hand, Lee-Topping [10] proved that nonnegativity of scalar curvature is not preserved in dimension at least four under the topology of uniform convergence of Riemannian distance. For other studies about the behaviour of the scalar curvature under a weak topology, see, for example, [3,6,7].…”

Rigidity of Lipschitz map using harmonic map heat flow

“…Suppose g is a C 0 metric. If g is satisfies one of the following, then it is known that it also has scalar curvature bounded below by σ in the sense of Definition 1.2: (a) g has scalar curvature bounded below by σ in the sense of Burkhardt-Guim [3] using regularizing Ricci flow; (b) g ∈ W 1,p , p > n with R ≥ σ in the sense of distribution as in [21] by Jiang-Sheng-Zhang [18] ; (c) g is smooth away from singularity Σ of co-dimension at least three and has R(g) ≥ σ outside Σ by the work [22] of the authors; (d) there exist smooth g i → g in C 0 norm so that some integral form of lower bound of R(g i ) is satisfied by Huang and the first named author [12].…”