On convergence to a football (Q330831)

The main goal of the paper is to show the following fact: Any sequence of Riemann spheres of positive constant curvature with more than 2 conic points passing from the stable to the semi-stable case has a subsequence which converges in the Gromov-Haudorff topology to the so-called football, i.e., the Riemann sphere with two conic points with conic singularity of the same order. From the geometrical point of view this fact can mean that all but one point \(s\) will merge into a single conic point of the limit sphere. The main tool of the proof are two results proposed by \textit{F. Luo} and \textit{G. Tian} [Proc. Am. Math. Soc. 116, No. 4, 1119--1129 (1992; Zbl 0806.53012)], the summing of cone angles among others.