Type II ancient compact solutions to the Yamabe flow

We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t -> -infinity, to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments, based on sharp estimates on ancient solutions of the approximated linear equation and careful estimation of the error terms which allow us to make the right choice of parameters. Our technique may be viewed as a parabolic analogue of gluing two exact solutions to the rescaled equation, that is the spheres, with narrow cylindrical necks to obtain a new ancient solution to the Yamabe flow. The result generalizes to the gluing of k spheres for any k >= 2, in such a way the configuration of radii of the spheres glued is driven as t -> -infinity by a First order Toda system.