Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2202.04941 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909345938145280 |
|---|---|
| author | Tschanz, Léonard |
| author_facet | Tschanz, Léonard |
| contents | We introduce a graph $Γ$ which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary $Ω$ of $Γ$. For $(Ω_l)_{l\geq 1}$ a sequence of subraphs of $Γ$ such that $|Ω_l| \longrightarrow \infty$, we prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B_l|$. The idea of the proof consists in finding a bounded domain $N$ of the hyperbolic plane which is roughly isometric to $Ω$, giving an upper bound for the Steklov eigenvalues of $N$ and transferring this bound to $Ω$ via a process called discretization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_04941 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Steklov problem on triangle-tiling graphs in the hyperbolic plane Tschanz, Léonard Differential Geometry We introduce a graph $Γ$ which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary $Ω$ of $Γ$. For $(Ω_l)_{l\geq 1}$ a sequence of subraphs of $Γ$ such that $|Ω_l| \longrightarrow \infty$, we prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B_l|$. The idea of the proof consists in finding a bounded domain $N$ of the hyperbolic plane which is roughly isometric to $Ω$, giving an upper bound for the Steklov eigenvalues of $N$ and transferring this bound to $Ω$ via a process called discretization. |
| title | The Steklov problem on triangle-tiling graphs in the hyperbolic plane |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2202.04941 |