Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature Disks

We compute the scalar determinants $\det (\mathrm{\Delta }+M^2)$ on the two-dimensional round disks of constant curvature $R=0$, $\mp 2$, for any finite boundary length $\mathrm{\ell }$ and mass $M$, with Dirichlet boundary conditions, using the $\zeta$-function prescription. When $M^2=\pm q(q+1)$, $q\in \mathbb{N}$, a simple expression involving only elementary functions and the Euler $\mathrm{\Gamma }$ function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.