Surfaces of constant curvature in R3 with isolated singularities

We prove that finite area isolated singularities of surfaces with constant positive curvature K>0 in R3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in S2 with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R3, i.e. compact convex surfaces of constant curvature K>0 with a finite number of singularities, and give applications to harmonic maps and constant mean curvature surfaces.