The zeta-determinant of the generalized Dirichlet-to-Neumann operator associated to the Alvarez boundary condition

초록

The gluing formula for the zeta-determinants of the Dolbeault Laplacians on a compact Riemann surface was proved by R. Wentworth by using the Alvarez boundary condition. The Alvarez boundary condition is defined by using the complex structure on a compact Riemann surface and in a case of a trivial line bundle it is reduced to the half Dirichlet and half Neumann boundary condition. He introduced a generalized Dirichlet-to-Neumann operator associated to the Alvarez boundary condition, which is a classical elliptic pseudodifferential operator of order zero, and used it to prove the gluing formula. In this talk, I’m going to describe the generalized Dirichlet-to-Neumann operator associated to the Alvares boundary condition on a trivial line bundle in terms of the classical Dirichlet-to-Neumann operators associated to the Dirichlet boundary condition and show that the gluing formula for the Dolbeault Laplacian is the same as the gluing formula for the Dirichlet and Neumann conditions.