标题： 构造具有任意大重数的第一Steklov特征值的曲面 显示英文标题

标题： Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity

摘要： 我们构造具有任意大的第一个非零斯蒂尔杰斯特征值重数的曲面。 证明基于M. Burger和B. Colbois提出的技术，该技术最初用于证明拉普拉斯谱的类似结果。 我们首先构造曲面$S_p$，对于每个素数$p$，其具有特定的等距同构子群$G_p:= \mathbb{Z}_p \rtimes \mathbb{Z}_p^*$。 我们通过按照$G_p$的凯莱图结构粘合带有边界的曲面来实现这一点。 我们随后利用$G_p$和$S_p$的性质，以证明高次数（取决于$p$）的不可约表示作用在与$\sigma_1(S_p)$相关的函数的特征空间上，从而得到所需的结果。 显示英文摘要

摘要： We construct surfaces with arbitrarily large multiplicity for their first non-zero Steklov eigenvalue. The proof is based on a technique by M. Burger and B. Colbois originally used to prove a similar result for the Laplacian spectrum. We start by constructing surfaces $S_p$ with a specific subgroup of isometry $G_p:= \mathbb{Z}_p \rtimes \mathbb{Z}_p^*$ for each prime $p$. We do so by gluing surfaces with boundary following the structure of the Cayley graph of $G_p$. We then exploit the properties of $G_p$ and $S_p$ in order to show that an irreducible representation of high degree (depending on $p$) acts on the eigenspace of functions associated with $\sigma_1(S_p)$, leading to the desired result.