Let $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

令 $\Sigma $ 为有边界的紧致曲面。 对于$\Sigma $ 上的给定保形类c ，函数 $\sigma _k^*(\Sigma ,c)$ 被定义为c中所有度量的第k个归一化 Steklov 特征值的上确界。我们考虑这个泛函在 $\Sigma $ 上的保形类的模空间上的行为。得到了序列 $\{c_n\}$ 退化时 $\sigma_k^*(\Sigma,c_n)$ 极限的精确公式。我们将这个公式应用于研究封闭流形的 Friedlander-Nadirashvili 不变量的自然类似物，定义为 $\inf _{c}\sigma _k^*(\Sigma ,c)$ ，其中下确界接管了 $\Sigma $ 上c。我们证明，对于任何有边界的表面，这些量等于 $2\pi k$ 。作为我们技术的应用，我们根据其属和边界分量的数量获得了关于不可k个归一化 Steklov 特征值的新估计。