Can One Perturb the Equatorial Zone on a Sphere with Larger Mean Curvature?

Abstract

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.