Ricci-DeTurck flow of almost continuous $L^2$-metrics, and metrics with distributional scalar curvature bounded from below
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918222299660288 |
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| author | Litzinger, Florian Simon, Miles |
| author_facet | Litzinger, Florian Simon, Miles |
| contents | 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 $t>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23234 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ricci-DeTurck flow of almost continuous $L^2$-metrics, and metrics with distributional scalar curvature bounded from below Litzinger, Florian Simon, Miles Differential Geometry Analysis of PDEs 53E20 (Primary) 47J35, 35K55 (Secondary) 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 $t>0$. |
| title | Ricci-DeTurck flow of almost continuous $L^2$-metrics, and metrics with distributional scalar curvature bounded from below |
| topic | Differential Geometry Analysis of PDEs 53E20 (Primary) 47J35, 35K55 (Secondary) |
| url | https://arxiv.org/abs/2511.23234 |