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- Title
Ricci flow of W <sup>2,2</sup>-metrics in four dimensions.
- Authors
Lamm, Tobias; Simon, Miles
- Abstract
In this paper we construct solutions to Ricci–DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values g are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to W2,2 and satisfy 1/a h ≤ g ≤ ah for some 1 < a < ∞ and some smooth Riemannian metric h on M. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci–DeTurck solution. Results for a related non-compact setting are also presented. Various Lp-estimates for Ricci flow, which we require for some of the main results, are also derived. As an application we present a possible definition of scalar curvature ≥ k for W2,2-metrics g on closed four manifolds which are bounded in the L∞-sense by 1/a h ≤ g ≤ ah for some 1 < a < ∞ and some smooth Riemannian metric h on M.
- Subjects
RICCI flow; RIEMANNIAN metric; CURVATURE
- Publication
Commentarii Mathematici Helvetici, 2023, Vol 98, Issue 2, p261
- ISSN
0010-2571
- Publication type
Article
- DOI
10.4171/CMH/553