# Small energy implies a lifting $\rho e^{i\theta}$

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!

• Only in dimension $1$. In higher dimensions you can create a function with arbitrarily small energy that vanishes on an open set, after which you can change it a bit on that open set to make the argument $\theta$ behaving in a bad way. Jun 1 '19 at 22:29
• But always, independently of the dimension, $|u|$ is close to $1$ if the energy is small, isnt it? The problem is that $\theta$ is not $H^1(T^N)$ when $N>1$. Am I right? Jun 2 '19 at 14:10
• Only on a big set. On a small set $|u|$ can be as far from $1$ as you want. Jun 2 '19 at 14:15