Paper · Sep 17, 2021

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

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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 $g$ are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to $W^{2, 2}$ and satisfy $\frac{1}{a} h \leq g \leq ah$ for some $1 < a < \infty$ 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 $L^{p}$ 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 $\geq k$ for $W^{2, 2}$ metrics $g$ on closed four manifolds which are bounded in the $L^{\infty}$ sense by $\frac{1}{a} h \leq g \leq ah$ for some $1 < a < \infty$ and some smooth Riemannian metric $h$ on $M$ .

Comments: minor corrections, to appear in Comment. Math. Helv

Ricci flow 4-manifolds Scalar curvature Weak & singular flows Boundaries & non-compact Harmonic maps & DeTurck trick

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Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

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

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Ricci flow of W2,2-metrics in four dimensions

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