We give local and global classification results for surfaces of positive constant curvature in $\mathbb{R}^3$ in the presence of isolated singularities. In particular, we will prove removable singularity theorems, we will describe the space of local immersions of constant curvature around a conical singularity, and we will provide some applications to harmonic maps and CMC surfaces of the classification of peaked spheres, i.e. compact convex surfaces of constant positive curvature with a finite number of singularities.
This is a joint work with J.A. Galvez and L. Hauswirth.