Blow-up analysis of a nonlocal Liouville-type equation

In this paper we establish an equivalence between the Nirenberg problem on the circle and the boundary of holomorphic immersions of the disk into the plane. More precisely we study the following nonlocal Liouville-type equation
\[
(1) (-\Delta)^{1/2} u = \kappa e^u - 1 in S^1
\]
where $(-\Delta)^{1/2}$ stands for the fractional Laplacian and $\kappa$ is a bounded function. The equation (1) can actually be interpreted as the prescribed curvature equation to a curve in conformal parametrization. Thanks to this geometric interpretation we perform a subtle blow-up and quantization analysis of (1). We also show a relation between equation (1) and the analogous equation in $\mathbb{R}$
\[
(2) (-\Delta)^{1/2} u = K e^u in \mathbb{R},
\]
with K bounded on $\mathbb{R}$.