On the best pinching constant of conformal metrics on S^2 with one and two conical singularities

We answer a long-standing open question asked by Thurston (The Geometry and Topology of Three-Manifolds. Princeton University Press, Princeton, 1978) concerning the best pinching constant for conformal metrics on S2 with one and two conical singularities of angles 2π(1+α 1) and 2π(1+α 1),2π(1+α 2) in case α 1∈(−1,0) and −1<α 1<α 2<0, respectively. The case of one conical singularity is a corollary of a result in Chen and Lin (Commun. Anal. Geom. 6(1):1–19, 1998) concerning the curvature of conformal metrics on ℝ2 with bounded Gaussian curvature 0<a≤K≤b<+∞. The case with two conical singularities is worked out by a generalization of that result.