A Liouville Theorem for Axi-symmetric Navier-Stokes Equations on ${\mathbb{R}}^2\times {\mathbb{T}}^1$

We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty, 0] \times ($\mathbb{R}$^2 \times $\mathbb{T}$^1)$. This is a step forward to completely solve the conjecture on $(-\mathrm{\infty },0]\times {\mathbb{R}}^3$ which was made in \cite{KNSS} to describe the potential singularity structures of the Cauchy problem.