Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.06713 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908822099984384 |
|---|---|
| author | Baer, Christian |
| author_facet | Baer, Christian |
| contents | Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin with a constant that also depends on a lower bound on the mean curvature $H$ (which is allowed to take negative values). If the boundary is a (not necessarily convex) hypersurface in a space form of non-negative curvature, then the constant can be made explicit in terms of the mean curvature of this model embedding. If the boundary has constant sectional curvature $κ>0$ and is a projective space of dimension $n\equiv 3 \mod 4$ or a sphere, then the constant can be expressed in terms of $κ$. If the boundary is a flat torus, then the constant can be expressed in terms of lattice data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06713 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Upper bound for the total mean curvature of spin fill-ins Baer, Christian Differential Geometry 53C20, 53C27 Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin with a constant that also depends on a lower bound on the mean curvature $H$ (which is allowed to take negative values). If the boundary is a (not necessarily convex) hypersurface in a space form of non-negative curvature, then the constant can be made explicit in terms of the mean curvature of this model embedding. If the boundary has constant sectional curvature $κ>0$ and is a projective space of dimension $n\equiv 3 \mod 4$ or a sphere, then the constant can be expressed in terms of $κ$. If the boundary is a flat torus, then the constant can be expressed in terms of lattice data. |
| title | Upper bound for the total mean curvature of spin fill-ins |
| topic | Differential Geometry 53C20, 53C27 |
| url | https://arxiv.org/abs/2601.06713 |