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We prove that any hyperbolic end with particles (cone singularities alonginfinite curves of angles less than $\pi$) admits a unique foliation byconstant Gauss curvature surfaces. Using a form of duality between hyperbolicends with particles and convex globally hyperbolic maximal (GHM) de Sitterspacetime with particles, it follows that any convex GHM de Sitter spacetimewith particles also admits a unique foliation by constant Gauss curvaturesurfaces. We prove that the grafting map from the product of Teichm\"ullerspace with the space of measured laminations to the space of complex projectivestructures is a homeomorphism for surfaces with cone singularities of anglesless than $\pi$, as well as an analogue when grafting is replaced by "smoothgrafting".

Hyperbolic ends with particles and grafting on singular surfaces