Geometry Seminar: M. Simon (Otto von Guericke University Magdeburg)
Geometry Seminar (Geometric Analysis)
Speaker: Miles Simon (Otto von Guericke University Magdeburg)
Title: Ricci-deturck flow of almost continuous $L^2$-metrics, and metrics with distributional scalar curvature bounded from below
Speaker: Miles Simon (Otto von Guericke University Magdeburg)
Title: Ricci-deturck flow of almost continuous $L^2$-metrics, and metrics with distributional scalar curvature bounded from below
Abstract: We consider Riemannian manifolds $\left(M^n, g_0\right),\left(M^n, h\right)$, where $\left(M^n, h\right)$ is smooth, complete, with curvature bounded in absolute value by $K_0<\infty$, and $\left(1-\varepsilon_0(n)\right) h \leq g_0 \leq\left(1+\varepsilon_0(n)\right) h$ 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 $T=T\left(n, K_0\right)>0$. If $g_0 \in L_{\text {loc }}^2$ or $g_0 \in W_{\text {loc }}^{1,2+2 \sigma}$, $\sigma \in\left(0, \frac{1}{4}\right)$, respectively, we show that $g(t) \rightarrow g_0$ in the $L_{\text {loc }}^2$ - or $W_{\text {loc }}^{1,2+\sigma}$-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 $t>0$. This is joint work with FLORIAN LITZINGER.