Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915191297409024 |
|---|---|
| author | Impera, Debora Pigola, Stefano Rimoldi, Michele Veronelli, Giona |
| author_facet | Impera, Debora Pigola, Stefano Rimoldi, Michele Veronelli, Giona |
| contents | We establish the validity of the isoperimetric inequality (or equivalently, an $L^1$ Euclidean-type Sobolev inequality) on manifolds with asymptotically non-negative sectional curvature. Unlike previous results in the literature, our approach does not require the negative part of the curvature to be globally small. Furthermore, we derive a Michael-Simon inequality on manifolds whose curvature is non-negative outside a compact set.
The proofs employ the ABP method for isoperimetry, initially introduced by Cabré in the Euclidean setting and subsequently extended and skillfully adapted by Brendle to the challenging context of non-negatively curved manifolds. Notably, we show that this technique can be localized to appropriate regions of the manifold. Additional key elements of the argument include the geometric structure at infinity of asymptotically non-negatively curved manifolds, their spectral properties - which ensure the non-negativity of a Bakry-Émery Ricci tensor on a conformal deformation of each end - and a result that deduces the validity of the isoperimetric inequality on the entire manifold, provided it holds outside a compact set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08279 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature Impera, Debora Pigola, Stefano Rimoldi, Michele Veronelli, Giona Differential Geometry Analysis of PDEs 53C21, 53C40 We establish the validity of the isoperimetric inequality (or equivalently, an $L^1$ Euclidean-type Sobolev inequality) on manifolds with asymptotically non-negative sectional curvature. Unlike previous results in the literature, our approach does not require the negative part of the curvature to be globally small. Furthermore, we derive a Michael-Simon inequality on manifolds whose curvature is non-negative outside a compact set. The proofs employ the ABP method for isoperimetry, initially introduced by Cabré in the Euclidean setting and subsequently extended and skillfully adapted by Brendle to the challenging context of non-negatively curved manifolds. Notably, we show that this technique can be localized to appropriate regions of the manifold. Additional key elements of the argument include the geometric structure at infinity of asymptotically non-negatively curved manifolds, their spectral properties - which ensure the non-negativity of a Bakry-Émery Ricci tensor on a conformal deformation of each end - and a result that deduces the validity of the isoperimetric inequality on the entire manifold, provided it holds outside a compact set. |
| title | Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature |
| topic | Differential Geometry Analysis of PDEs 53C21, 53C40 |
| url | https://arxiv.org/abs/2503.08279 |