Massive Cantor families of periodic solutions of resonant Klein-Gordon equation on ${\mathbb{S}}^3$

We prove existence and multiplicity of Cantor families of small amplitude analytic in time periodic solutions of the completely resonant cubic nonlinear Klein-Gordon equation on ${\mathbb{S}}^3$ for an asymptotically full measure set of frequencies close to 1. The solutions are constructed by a Lyapunov-Schmidt decomposition and a Nash-Moser iterative scheme. We first find non-degenerate solutions of the Kernel equation. Then we solve the Range equation with a Nash-Moser iterative scheme to overcome small divisors problems.