Degenerating conic Kähler-Einstein metrics to the normal cone

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1α(-K_X)$ with $α>1$. It is well-known that Kähler-Einstein metrics on $X$ with conic singularities along $D$ may exist only if the angle $2πβ$ is bigger than some positive limit value $2πβ_*$. Under the hypothesis that the automorphisms of $D$ are induced by the automorphisms of the pair $(X,D)$, we prove that for $β>β_*$ close enough to $β_*$, such Kähler-Einstein metrics do exist. We identify the limits at various scales when $β\rightarrowβ_*$ and, in particular, we exhibit the appearance of the Tian-Yau metric of $X\setminus D$.