K\"ahler-Einstein metrics: from cones to cusps

In this note, we prove that on a compact K\"ahler manifold $X$ carrying a smooth divisor $D$ such that $K_X+D$ is ample, the K\"ahler-Einstein cusp metric is the limit (in a strong sense) of the K\"ahler-Einstein conic metrics when the cone angle goes to $0$. We further investigate the boundary behavior of those and prove that the rescaled metrics converge to a cylindrical metric on $\mathbb C^*\times \mathbb C^{n-1}$.