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

TL;DR

This paper advances the understanding of the third gap problem in Simon's conjecture for minimal surfaces in spheres by developing new integral identities and establishing positive gap results in a specific curvature interval.

Contribution

It introduces refined third-order Simons-type identities and new lower bounds, extending gap results to the entire interval including endpoints.

Findings

Established positive gap results for the squared norm of the second fundamental form in [5/3, 9/5].

Developed refined third-order Simons-type integral identities.

Proved a rigidity result for closed self-shrinkers.

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.