The Steklov problem on triangle-tiling graphs in the hyperbolic plane

Publication date: 10 February 2022

Abstract: We introduce a graph

G a m m a

which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary

O m e g a

of

G a m m a

. For

(O m e g a_{l})_{l g e q 1}

a sequence of subraphs of

G a m m a

such that

| O m e g a_{l} | l o n g r i g h t a r r o w i n f t y

, we prove that for each

k i n m a t h b b N

, the

k^{m b o x t h}

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

O m e g a

, giving an upper bound for the Steklov eigenvalues of

N

and transferring this bound to

O m e g a

via a process called discretization.

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