*preprint*

**Inserted:** 17 jul 2018

**Year:** 2015

**Abstract:**

In this paper we perform a blow-up and quantization analysis of the following
nonlocal Liouville-type equation \begin{equation}(-\Delta)^{\frac12} u= \kappa
e^{u}-1~\mbox{in $S^1$,} \end{equation} where $(-\Delta)^\frac{1}{2}$ stands for
the fractional Laplacian and $\kappa$ is a bounded function. We interpret the
above equation as the prescribed curvature equation to a curve in conformal
parametrization. We also establish a relation between this equation and the
analogous equation in $\mathbb{R}$ \begin{equation}
(-\Delta)^{\frac{1}{2}} u =Ke^{u} \quad \text{in }\mathbb{R}, \end{equation} with
$K$ bounded on $\mathbb{R}$.