Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature Disks

We compute exactly the scalar determinants (Δ+M²) on the two-dimensional round disks of constant curvature $R=0$, ∓ 2, for any finite boundary length and mass M, with Dirichlet boundary conditions, using the ζ-function prescription. When M²=± q(q+1), q∈ N, a simple expression involving only elementary functions and the Euler Γ function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.