Existence of minimal maps of degree one in $W^{\frac{1}{p},p}({\mathbb{S}}^1,{\mathbb{S}}^1)$ for $p\in [p^{\mathrm{\prime }},2]$, where $p^{\mathrm{\prime }}\approx 1.13924$

In this note, we show how the results of Mazowiecka–Schikorra, combined with those of Bourgain–Brezis–Mironescu, imply the existence of minimal maps of degree one in $W^{\frac{1}{p},p}({\mathbb{S}}^1,{\mathbb{S}}^1)$ for $p\in [p^{\mathrm{\prime }},2]$, where $p^{\mathrm{\prime }}\approx 1.13924$. This provides an affirmative answer in this range to a question posed by Mironescu and Brezis–Mironescu. In order to do so, we complement the results of Mazowiecka–Schikorra by extending them to the case $n=1$ and $1<p<2$, which had been excluded there for technical reasons.