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

Abstract: We consider Riemannian manifolds $(M<sup>n,g_0)$, $(M<sup>n,h)$, where $(M<sup>n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0 &lt; \infty$, and $$(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$$ for some small $\varepsilon_0(n)&gt;0$. It was shown by Simon (2002) that a Ricci-DeTurck flow solution $g(t)<em>{t \in (0,T)}$ related to $g_0$ exists for some $T=T(n,K_0)&gt;0$. If $g_0 \in L²</em>{\mathrm{loc}}$ or $g_0 \in W<sup>{1,2+2σ}_{\mathrm{loc}}$, $σ\in (0,\frac{1}{4})$, respectively, we show that $g(t) \to g_0$ in the $L<sup>2_{\mathrm{loc}}$- or $W<sup>{1,2+σ}_{\mathrm{loc}}$-sense, respectively. If $M$ is closed, $g_0 \in W<sup>{1,2+σ}(M)$ for some $σ&gt;0$, 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 $t&gt;0$.