Ricci-DeTurck flow of almost continuous $L^2$-metrics, and metrics with distributional scalar curvature bounded from below
Florian Litzinger, Miles Simon
TL;DR
The paper extends the theory of Ricci--DeTurck flow to initial data that are almost continuous and belong to $L^2_{\mathrm{loc}}$ or $W^{1,2+2\sigma}_{\mathrm{loc}}$, establishing local convergence of the flow to the initial metric and providing rate and Sobolev estimates. It shows that, for closed manifolds and distributional lower bounds on scalar curvature, the flow preserves these lower bounds in the smooth sense for $t>0$ under appropriate regularity assumptions, using the Lee--LeFloch distributional framework and conjugate heat equation techniques. The results include uniform-in-time $W^{1,2+2\sigma}_{\mathrm{loc}}$ bounds and convergence in $W^{1,2+\sigma}_{\mathrm{loc}}$ for $0\le\sigma<\tfrac{1}{4}$, with detailed energy estimates built around a weighted gradient quantity. Overall, the work provides smoothing and curvature-preservation results for non-smooth initial data via Ricci--DeTurck flow and its related Ricci flow, contributing to the analysis of metrics with distributional curvature bounds.
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 $g_0 \in W^{1,2+2σ}_{\mathrm{loc}}$, $σ\in (0,\frac{1}{4})$, respectively, we show that $g(t) \to g_0$ in the $L^2_{\mathrm{loc}}$- or $W^{1,2+σ}_{\mathrm{loc}}$-sense, respectively. If $M$ is closed, $g_0 \in W^{1,2+σ}(M)$ for some $σ>0$, 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 $t>0$.