  We find out the complex geometries corresponding to the semi-classical
saddles of threedimensional quantum gravity by making use of the known results
of dual conformal field theory (CFT), which is effectively given by Liouville
field theory. We examine both the cases with positive and negative cosmological
constants. We determine the set of semi-classical saddles to choose from the
homotopy argument in the Chern-Simons formulation combined with CFT results and
provide strong supports from the mini-superspace approach to the quantum
gravity. For the case of positive cosmological constant, partial results were
already obtained in our previous works, and they are consistent with the
current ones. For the case of negative cosmological constant, we identify the
geometry corresponding a semi-classical saddle with three-dimensional Euclidean
anti-de Sitter space dressed with imaginary radius three-dimensional spheres.
The geometry is generically unphysical, but we argue that the fact itself does
not lead to any problems.
