Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913468475506688 |
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| author | Burkhardt-Guim, Paula |
| author_facet | Burkhardt-Guim, Paula |
| contents | We show that any $L^\infty$ Riemannian metric $g$ on $\mathbb{R}^n$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-α)$-dimensional Minkowski content, for some $α>2$, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that $g$ is sufficiently close in $L^\infty$ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in $C^\infty$ to $g$ away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a $L^\infty$ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set Burkhardt-Guim, Paula Differential Geometry 53E20, 53C21 We show that any $L^\infty$ Riemannian metric $g$ on $\mathbb{R}^n$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-α)$-dimensional Minkowski content, for some $α>2$, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that $g$ is sufficiently close in $L^\infty$ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in $C^\infty$ to $g$ away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a $L^\infty$ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times. |
| title | Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set |
| topic | Differential Geometry 53E20, 53C21 |
| url | https://arxiv.org/abs/2406.04564 |