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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!

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    $\begingroup$ 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. $\endgroup$
    – fedja
    Jun 1 '19 at 22:29
  • $\begingroup$ 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? $\endgroup$ Jun 2 '19 at 14:10
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    $\begingroup$ Only on a big set. On a small set $|u|$ can be as far from $1$ as you want. $\endgroup$
    – fedja
    Jun 2 '19 at 14:15

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