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 $(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$ for some small $\varepsilon_0(n)>0$. 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 $g_0 \in L^2_{\mathrm{loc}}$ or $g_0 \in W^{1,2+2\sigma}_{\mathrm{loc}}$, $\sigma \in (0,\frac{1}{4})$, respectively, we show that $g(t) \to g_0$ in the $L^2_{\mathrm{loc}}$- or $W^{1,2+\sigma}_{\mathrm{loc}}$-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 $t>0$.