AI 中文总结
本文通过Lee-Uhlmann全符号公式与Weyl不变量理论计算Dirichlet-to-Neumann算子的Guillemin-Wodzicki留数，进而证明三维空间形式中测地球及同心球壳区域的Steklov谱唯一性与刚性定理。
AI 中文摘要
设Λ为紧致三维黎曼流形边界上的Dirichlet-to-Neumann算子，本文利用Lee-Uhlmann全符号公式与Weyl不变量理论，显式计算Λ与Λ²的Guillemin-Wodzicki留数，进而得到Steklov热迹的前两个对数系数。作为应用，证明单连通三维空间形式中的测地球，在同空间形式的光滑区域中由其Steklov谱唯一确定；更一般地，得到具有光滑（不一定连通）边界的紧致常曲率流形类中的刚性定理；在欧氏情形下，还证明了同心球壳区域的谱唯一性。
英文摘要
Let $Λ$ be the Dirichlet-to-Neumann operator on the boundary of a compact three-dimensional Riemannian manifold. Using the Lee--Uhlmann full-symbol formula and Weyl's invariant theory, we compute the Guillemin--Wodzicki residues of $Λ$ and $Λ^2$ explicitly, and hence the first two logarithmic coefficients in the Steklov heat trace. As an application, we prove that geodesic balls in simply connected three-dimensional space forms are determined by their Steklov spectra among smooth domains in the same space form. More generally, we obtain a rigidity theorem in the class of compact constant-curvature manifolds with smooth, not necessarily connected, boundary. In the Euclidean case, we also prove spectral uniqueness for concentric spherical shell regions.
