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arXiv · 2210.00994

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

Baichuan Hu·Xiang Ma·Shengyang Wang
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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.

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Baichuan Hu, Xiang Ma, Shengyang Wang. 2022-10-03. Can One Perturb the Equatorial Zone on a Sphere with Larger Mean Curvature?. https://arxiv.org/abs/2210.00994

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