Uniformization of surfaces with boundary and the application to the triple junction surfaces with negative Euler characteristic

The following is a short video shows that why the uniqueness might fail when $K>0$.

In this video, we suppose the surface $\Sigma$ is the two copies of the domain enclosed by six arcs by identifying their corresponding edges marked with green color. So here $\Sigma$ is just a genus zero surface with three boundary components. We fix one boundary of $\Sigma$ having geodesic curvature $-1$ (marked as red) and the remaining two boundaries being geodesic (marked as black). Then we can let the area grow to see how the Gauss curvature changes after uniformization. The blue circle in this video represents the infinity circle if $K<0$ or the geodesic circle if $K>0$. From the video, we can see that $K$ will increase as area $A$ increases, then $K$ will reach its maximal point and after that $K$ will decrease to 0. So we will see that for some $K>0$, there are indeed two metrics with prescribed Gauss curvature and geodesic curvatures of boundaries.