The determinant of the Dirichlet-to-Neumann map for surfaces with boundary

Colin Guillarmou (1) , Laurent Guillopé (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 LMJL - Laboratoire de Mathématiques Jean Leray

For any orientable compact surface with boundary, we compute the regularized determinant of the Dirichlet-to-Neumann (DN) map in terms of main value at 0 of a Ruelle zeta function using uniformization of Mazzeo-Taylor. We apply it to compact hyperbolic surfaces with totally geodesic boundary and obtain various interpretations of the determinant of the Dirichlet Laplacian in terms of dynamical zeta functions. We also relate in any dimension the DN map for the Yamabe operator to the scattering operator for a conformally compact related problem by using uniformization.

Domaines: Théorie spectrale [math.SP]

Colin Guillarmou, Laurent Guillopé. The determinant of the Dirichlet-to-Neumann map for surfaces with boundary. 2007. ⟨hal-00126393v1⟩