Spectral theory of singular surfaces

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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.