论文标题
共形类别的退化序列和共形steklov谱
Degenerating sequences of conformal classes and the conformal Steklov spectrum
论文作者
论文摘要
令$σ$为具有边界的紧凑型表面。对于$σ$上的给定的共形级$ c $，功能$σ_k^*（σ，c）$定义为$ k- $ th标准化的steklov eigenvalue的最高指标。我们考虑了此功能在$σ$上共形类的模量空间上的行为。 $σ_k^*（σ，c_n）$的精确公式当获得序列$ \ {c_n \} $变性时。我们将此公式应用于封闭歧管的Friedlander-nadirashvili不变的自然类似物，定义为$ \ inf_ {c}σ_k^*（σ_k^*（σ，c）$，其中$ c $ in $σ$ in $ c $。我们证明，对于任何具有边界的表面，这些数量等于$2πk$。作为我们技术的应用，我们获得了关于非定向表面的$ k $ th标准化的steklov特征值的新估计，其属和边界成分的数量。
Let $Σ$ be a compact surface with boundary. For a given conformal class $c$ on $Σ$ the functional $σ_k^*(Σ,c)$ is defined as the supremum of the $k-$th normalized Steklov eigenvalue over all metrics on $c$. We consider the behaviour of this functional on the moduli space of conformal classes on $Σ$. A precise formula for the limit of $σ_k^*(Σ,c_n)$ 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 $\inf_{c}σ_k^*(Σ,c)$, where the infimum is taken over all conformal classes $c$ on $Σ$. We show that these quantities are equal to $2πk$ for any surface with boundary. As an application of our techniques we obtain new estimates on the $k-$th normalized Steklov eigenvalue of a non-orientable surface in terms of its genus and the number of boundary components.
