Ricci-DeTurck flow of almost continuous $L^2$-metrics, and metrics with distributional scalar curvature bounded from below

Abstract

We consider Riemannian manifolds $(M^n,g_0)$, $(M^n,h)$, where $(M^n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0<\infty$, and $(1-{\epsilon }_0(n))h\le g_0\le (1+{\epsilon }_0(n))h$ for some small ${\epsilon }_0(n)>0$. It was shown by Simon (2002) that a Ricci-DeTurck flow solution $g(t{)}_{t\in (0,T)}$ related to $g_0$ exists for some $T=T(n,K_0)>0$. If $g_0\in L_{\mathrm{loc}}^2$ or $g_0\in W_{\mathrm{loc}}^{1,2+2\sigma }$, $\sigma \in (0,\frac{1}{4})$, respectively, we show that $g(t)\to g_0$ in the $L_{\mathrm{loc}}^2$- or $W_{\mathrm{loc}}^{1,2+\sigma }$-sense, respectively. If $M$ is closed, $g_0\in W^{1,2+\sigma }(M)$ for some $\sigma >0$, and the distributional scalar curvature of Lee-LeFloch (2015) is not less than $b\in \mathbb{R}$, then we show that $g(t)$ has scalar curvature not less than $b$ in the smooth sense for all $t>0$.