Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913520446078976 |
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| author | Eichmair, Michael Koerber, Thomas |
| author_facet | Eichmair, Michael Koerber, Thomas |
| contents | Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ with positive mass. We give a short proof based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of $(M, g)$ by stable constant mean curvature spheres. Moreover, we show that the geometric center of mass of the foliation agrees with the Hamiltonian center of mass of $(M,g)$. In dimension $n = 3$, these results were shown previously by C. Nerz using a different approach. In the case where $n=3$ and the scalar curvature of $(M, g)$ is nonnegative, we prove that the leaves of the asymptotic foliation are the only large stable constant mean curvature spheres that enclose the center of $(M, g)$. This was shown previously under more restrictive decay assumptions and using a different method by S. Ma. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_12081 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres Eichmair, Michael Koerber, Thomas Differential Geometry Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ with positive mass. We give a short proof based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of $(M, g)$ by stable constant mean curvature spheres. Moreover, we show that the geometric center of mass of the foliation agrees with the Hamiltonian center of mass of $(M,g)$. In dimension $n = 3$, these results were shown previously by C. Nerz using a different approach. In the case where $n=3$ and the scalar curvature of $(M, g)$ is nonnegative, we prove that the leaves of the asymptotic foliation are the only large stable constant mean curvature spheres that enclose the center of $(M, g)$. This was shown previously under more restrictive decay assumptions and using a different method by S. Ma. |
| title | Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2201.12081 |