Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere

We deduce an explicit closed formula for the zeta-regularized spectral determinant of the Friedrichs Laplacian on the Riemann sphere equipped with arbitrary constant curvature (flat, spherical, or hyperbolic) metric having three conical singularities of order $\beta_j\in(-1,0)$ (or, equivalently, of angle $2\pi(\beta_j+1)$). We show that among the metrics with a fixed value of the sum $\beta_1+\beta_2+\beta_3$ and a fixed surface area, those with $\beta_1=\beta_2=\beta_3$ correspond to a stationary point of the determinant. If, in addition, the surface area is sufficiently small, then the stationary point is a minimum. As a crucial step towards obtaining these results we find a relation between the determinant of Laplacian and the Liouville action introduced by A. Zamolodchikov and Al. Zamolodchikov in connection with the celebrated DOZZ formula for the three-point structure constants of the Liouville field theory.