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Ricci flow of $W^{2,2}$-metrics in four dimensions

Lamm, Tobias 1; Simon, Miles
1 Institut für Analysis (IANA), Karlsruher Institut für Technologie (KIT)

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 $W^{2,2}$ and satisfy $\frac{1}{a}$h≤g≤aha1​h≤g≤ah for some 1<a<∞ and some smooth Riemann\-ian 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 L$^p$-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≥k for $W^{2,2}$-metrics g on closed four manifolds which are bounded in the L$^∞$-sense by $\frac{1}{a}$h≤g≤ah for some 1<a<∞ and some smooth Riemannian metric h on M.


Verlagsausgabe §
DOI: 10.5445/IR/1000163895
Veröffentlicht am 13.11.2023
Originalveröffentlichung
DOI: 10.4171/CMH/553
Scopus
Zitationen: 1
Dimensions
Zitationen: 2
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2023
Sprache Englisch
Identifikator ISSN: 0010-2571, 1420-8946
KITopen-ID: 1000163895
Erschienen in Commentarii Mathematici Helvetici
Verlag European Mathematical Society
Band 98
Heft 2
Seiten 261 – 364
Vorab online veröffentlicht am 08.09.2023
Nachgewiesen in Scopus
Dimensions
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