On global smooth small data solutions of 3-D quasilinear Klein-Gordon equations on $ \mathbb{R}^{2}\times \mathbb{T} $

In this paper, for the small initial data in the suitably weighted Sobolev spaces on the product space $ \mathbb{R}^2\times\mathbb{T} $, we establish the global existence of smooth solutions to the 3-D quasilinear Klein-Gordon equations with quadratic nonlinearities. It is noted that the topic on the well-posedness of the nonlinear hyperbolic equations on product spaces arises from the studies for the propagation of waves along infinite homogeneous waveguides and the Kaluza-Klein theory. Our main result is based on the method of normal form including the parameters $ n\in\Bbb Z $ and the continuous induction method. In addition, the free profile of the solution is obtained by applying the weighted energy estimates and the temporal decay estimates.