  A spacelike surface with cylinder topology can be described by various sets
of canonical variables within pure AdS3 gravity. Each is made of one real
coordinate and one real momentum. The Hamiltonian can be either $H=0$ or it can
be nonzero and we display the canonical transformations that map one into the
other, in two relevant cases. In a choice of canonical coordinates, one of them
is the cylinder aspect $q$, which evolves nontrivially in time. The time
dependence of the aspect is an analytic function of time $t$ and an "angular
momentum" $J$. By analytic continuation in both $t$ and $J$ we obtain a
Euclidean evolution that can be described geometrically as the motion of a
cylinder inside the region of the 3D hyperbolic space bounded by two "domes"
(i.e. half spheres), which is topologically a solid torus. We find that for a
given $J$ the Euclidean evolution cannot connect an initial aspect to an
arbitrary final aspect; moreover, there are infinitely many Euclidean
trajectories that connect any two allowed initial and final aspects. We compute
the transition amplitude in two independent ways; first by solving exactly the
time-dependent Schr\"odinger equation, then by summing in a sensible way all
the saddle contributions, and we discuss why both approaches are mutually
consistent.
