环形曲面的斯捷克洛夫谱几何：逆谱结果与等谱紧致性
Steklov Spectral Geometry for Annular Surfaces: Inverse spectral results and isospectral compactness
AI总结：
研究带边界紧致曲面上斯捷克洛夫谱的逆谱问题，通过分析谱泽塔函数等，证明截头圆锥侧面由斯捷克洛夫谱唯一确定，圆形环域也如此，且斯捷克洛夫等谱平坦环形曲面族在\(C^\infty\)拓扑中紧致。
AI中文摘要：
我们研究了带边界紧致曲面上斯捷克洛夫谱的逆谱问题。证明了在平坦环形曲面中，截头圆锥的侧面由其斯捷克洛夫谱唯一确定。由此，每个圆形环域在所有平面区域中是唯一确定的，这是具有此性质的非单连通欧几里得区域的首个例子。此外，我们表明任何斯捷克洛夫等谱平坦环形曲面族在\(C^\infty\)拓扑中是紧致的，扩展了之前关于单连通平面区域的结果。这些结果是通过对与环形曲面狄利克雷 - 诺伊曼算子相关的谱泽塔函数和泽塔正则化行列式进行详细分析而建立的。
英文摘要：
We study the inverse spectral problems for the Steklov spectrum on compact surfaces with boundary. We prove that among flat annular surfaces, the lateral surface of a conical frustum is uniquely determined by its Steklov spectrum. As a consequence, each circular annulus is uniquely determined among all planar domains, providing the first example of a non-simply connected Euclidean domain with this property. Furthermore, we show that any family of Steklov isospectral flat annular surfaces is compact in the $C^\infty$ topology, extending previous results for simply connected planar domains. These results are established through a detailed analysis of the spectral zeta function and the zeta-regularized determinant associated with the Dirichlet-to-Neumann operator for annular surfaces.
