Ricci flow of -metrics in four dimensions
Tobias Lamm
Karlsruhe Institute of Technology (KIT), GermanyMiles Simon
Universität Magdeburg, Germany
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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 are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to and satisfy for some and some smooth Riemann\-ian metric on . 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 -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 for -metrics on closed four manifolds which are bounded in the -sense by for some and some smooth Riemannian metric on .
Cite this article
Tobias Lamm, Miles Simon, Ricci flow of -metrics in four dimensions. Comment. Math. Helv. 98 (2023), no. 2, pp. 261–364
DOI 10.4171/CMH/553