Spectral theory of singular surfaces

本课题致力于研究具有奇异点的常曲率曲面的谱理论，特别是具有锥形、圆柱形或欧几里得端点的平面和具有锥形点的常正曲率曲面。这些曲面提供了一个重要的实例，可以检验和改进一般的分析奇异流形的方法和散射理论。***我们项目的主要目标之一是研究恒定曲率奇异度量中标量和旋量拉普拉斯的各种谱特征（ζ函数的特殊值、个别特征值、行列式、散射矩阵等）与底层黎曼曲面模的关系。另一个目的是研究这些谱特征与选择对称拉普拉斯算子的自伴随扩展的关系。我们也计划研究频谱特征的极值问题（主要是拉普拉斯的行列式）被认为是模空间上的泛函。***本项目的结果可能用于获得关于各种模空间（紧黎曼曲面的模空间、亚纯函数的模的Hurwitz空间、亚纯微分的模空间等）的几何结构的重要信息；值得注意的是，奇异表面上拉普拉斯行列式的结果现在被研究量子霍尔效应的理论物理学家广泛使用，我们的研究部分是由这些应用驱动的。* * *

The project is devoted to the study of the spectral theory of surfaces of constant curvature with singularities, in particular, flat surfaces with conical, cylindrical or Euclidean ends and surfaces of constant positive curvature with conical points. These surfaces provide a source of nontrivial examples, where the general methods of the analysis on manifolds with singularities and the scattering theory can be tested and improved. *** Among the main objectives of our project is the investigation of the relation of various spectral characteristics of the scalar and spinor Laplacians in singular metrics of constant curvature (special values of zeta-function, individual eigenvalues, determinant, scattering matrix, etc) to the moduli of the underlying Riemann surface. Another goal is to study the dependence of these spectral characteristics on the choice of a self-adjoint extension of the symmetric Laplacian. We are also planning to study the extremal problems for spectral characteristics (mostly the determinant of the Laplacian) considered as functionals on the moduli spaces. ***Potentially, the results of our project can be used to get important information about the geometric structure of various moduli spaces (the moduli space of compact Riemann surfaces, the Hurwitz space of moduli of meromorphic functions, the moduli space of meromorphic differentials, etc); it should be noted that the results on the determinants of Laplacians on singular surfaces are now widely used by theoretical physicists working over quantum Hall effect, our research is partially motivated by these applications. ***