  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.
