Lower bounds for the low Steklov eigenvalues
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917544705654784 |
|---|---|
| author | Chakradhar, Tirumala Colbois, Bruno Hassannezhad, Asma |
| author_facet | Chakradhar, Tirumala Colbois, Bruno Hassannezhad, Asma |
| contents | For a compact, connected, orientable Riemannian manifold with $b$ boundary components, we obtain geometric lower bounds for the low Steklov eigenvalues, namely $σ_k$, $1\le k\le b-1$. Our results complement earlier results, which apply only to $σ_k$ with $k\ge b$ and depend on the geometry near the boundary, by showing how the interior geometry influences the low eigenvalues. Our result also yields lower bounds for the low Steklov eigenvalues in the setting of pinched negatively curved manifolds, thus recovering similar results in that context through an alternative proof. The proof of the main result is based on the trace inequality relating the Steklov eigenvalue to the Neumann eigenvalues of the connected subdomains of the manifold containing a boundary collar. The geometric coefficient appearing in this inequality is given by an explicit formula in terms of a quantity that can be interpreted as the electrical resistance of the boundary collar. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30254 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lower bounds for the low Steklov eigenvalues Chakradhar, Tirumala Colbois, Bruno Hassannezhad, Asma Differential Geometry Spectral Theory 58J50, 58C40, 35P15 For a compact, connected, orientable Riemannian manifold with $b$ boundary components, we obtain geometric lower bounds for the low Steklov eigenvalues, namely $σ_k$, $1\le k\le b-1$. Our results complement earlier results, which apply only to $σ_k$ with $k\ge b$ and depend on the geometry near the boundary, by showing how the interior geometry influences the low eigenvalues. Our result also yields lower bounds for the low Steklov eigenvalues in the setting of pinched negatively curved manifolds, thus recovering similar results in that context through an alternative proof. The proof of the main result is based on the trace inequality relating the Steklov eigenvalue to the Neumann eigenvalues of the connected subdomains of the manifold containing a boundary collar. The geometric coefficient appearing in this inequality is given by an explicit formula in terms of a quantity that can be interpreted as the electrical resistance of the boundary collar. |
| title | Lower bounds for the low Steklov eigenvalues |
| topic | Differential Geometry Spectral Theory 58J50, 58C40, 35P15 |
| url | https://arxiv.org/abs/2605.30254 |