A geometric boundary for the moduli space of grafted surfaces

Let $S$ be a closed orientable surface of genus at least 2. We consider three classes of geometric structures on $S$: hyperbolic metrics, singular flat metrics arising from half-translation surfaces, and those obtained via grafting — an operation introduced by Thurston to study complex projective structures. We show that these three families of metrics can be unified within a single connected moduli space. In particular, we prove that half-translation surfaces arise as geometric limits (up to rescaling) of grafted surfaces. Our proof relies on recent results on the orthogeodesic foliation constructed by Calderon and Farre.