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arXiv · 2603.03070

On Simon's third gap conjecture for minimal surfaces in spheres

Weiran Ding·Jianquan Ge·Fagui Li
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Abstract

In this paper, continuing our previous work, we investigate the third gap problem in the Simon conjecture for closed minimal surfaces in the unit sphere. By developing refined third-order Simons-type integral identities and establishing new lower bounds for higher-order curvature terms, we obtain positive gap results throughout the entire interval $\left[\frac{5}{3},\frac{9}{5}\right]$ for the squared norm of the second fundamental form, including the endpoint cases. As an application, we establish a rigidity result for closed self-shrinkers.

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Weiran Ding, Jianquan Ge, Fagui Li. 2026-03-03. On Simon's third gap conjecture for minimal surfaces in spheres. https://arxiv.org/abs/2603.03070

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