Pinching rigidity of minimal surfaces in spheres

In 1980, Simon proposed a quantization conjecture about the Gaussian curvature K of closed minimal surfaces in unit spheres: if K(s + 1) ⩽ K ⩽ K(s) (K(s) ≔ 2/(s(s + 1)), s ∈ ℕ), then either K = K(s) or K = K(s + 1). Notice that the surface must be one of Calabi’s standard minimal 2-spheres if the curvature is a positive constant. The cases s = 1 and s = 2 were proven in the 1980s by Simon and others. In this paper, we give a pinching theorem of the Simon conjecture in the case s = 3 and also give a new proof of the cases s = 1 and s = 2 by some Simons-type integral inequalities.