Small energy implies the existence of a lifting?

Question

Set $T^N$ the $N$-dimensional torus and $u\in H^1(T^N,\mathbb{C})$. Can I say that if the energy$$\int_{T^N}|\nabla u|^2 +\frac12\int_{T^N}[1-|u|^2]^2$$ is small enough (let say lower than some $\epsilon>0$), then $|u|$ is close to one, and therefore $u$ admits a lifting $u=\rho e^{i\theta}$ on the torus?. In that case, when can I assure that $\theta\in H^1(T^N,\mathbb{C})$. Any idea or comment is welcome. Thanks in advance!