Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929397217361920 |
|---|---|
| author | He, Shihang Shi, Yuguang Yu, Haobin |
| author_facet | He, Shihang Shi, Yuguang Yu, Haobin |
| contents | In this paper, we demonstrate that any asymptotically flat manifold $(M^n, g)$ with $4\leq n\leq 7$ can be foliated by a family of area-minimizing hypersurfaces, each of which is asymptotic to Cartesian coordinate hyperplanes defined at an end of $(M^n, g)$. As an application of this foliation, we show that for any asymptotically flat manifold $(M^n, g)$ with $4\leq n\leq 7$, nonnegative scalar curvature and positive mass, the solution of free boundary problem for area-minimizing hypersurface in coordinate cylinder $C_{R_i}$ in $(M^n, g)$ either does not exist or drifts to infinity of $(M^n, g)$ as $R_i$ tends to infinity. Additionally, we introduce a concept of globally minimizing hypersurface in $(M^n, g)$, and verify a version of the Schoen Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16242 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture He, Shihang Shi, Yuguang Yu, Haobin Differential Geometry In this paper, we demonstrate that any asymptotically flat manifold $(M^n, g)$ with $4\leq n\leq 7$ can be foliated by a family of area-minimizing hypersurfaces, each of which is asymptotic to Cartesian coordinate hyperplanes defined at an end of $(M^n, g)$. As an application of this foliation, we show that for any asymptotically flat manifold $(M^n, g)$ with $4\leq n\leq 7$, nonnegative scalar curvature and positive mass, the solution of free boundary problem for area-minimizing hypersurface in coordinate cylinder $C_{R_i}$ in $(M^n, g)$ either does not exist or drifts to infinity of $(M^n, g)$ as $R_i$ tends to infinity. Additionally, we introduce a concept of globally minimizing hypersurface in $(M^n, g)$, and verify a version of the Schoen Conjecture. |
| title | Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.16242 |