  We consider the mean curvature rigidity problem of an equatorial zone on a
sphere which is symmetric about the equator with width $2w$. There are two
different notions on rigidity, i.e. strong rigidity and local rigidity. We
prove that for each kind of these rigidity problems, there exists a critical
value such that the rigidity holds true if, and only if, the zone width is
smaller than that value. For the rigidity part, we used the tangency principle
and a specific lemma (the trap-slice lemma we established before). For the
non-rigidity part, we construct the nontrivial perturbations by a gluing
procedure called the round-corner lemma using the Delaunay surfaces.
