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

TL;DR

This paper investigates the rigidity of equatorial zones on spheres with larger mean curvature, establishing critical widths for rigidity and constructing perturbations for non-rigidity using geometric techniques.

Contribution

It introduces a critical width criterion for rigidity and develops a novel gluing method to construct nontrivial perturbations of the zone.

Findings

Existence of a critical width for rigidity and non-rigidity.

Rigidity holds if zone width is below the critical value.

Non-rigidity demonstrated through explicit perturbations.

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.