Gehring Link Problem, Focal Radius and Over-torical width

TL;DR

This paper investigates the Gehring link problem in the sphere, establishing a sphere theorem based on focal radius and confirming Gromov's conjectures in three dimensions.

Contribution

It introduces a sphere theorem for hypersurfaces in round spheres using focal radius and proves rigidity of Clifford hypersurfaces, confirming Gromov's conjectures in 3D.

Findings

Sphere theorem for hypersurfaces in S^n based on focal radius

Rigidity result for Clifford hypersurfaces in S^n

Confirmation of Gromov's conjectures in 3D

Abstract

In this note, we study the Gehring link problem in the round sphere, which motives our study of the width of a band in positively curved manifolds. Using the same idea, we are able to get a sphere theorem for hypersurface in in the round $S^{n}$ in terms of its focal radius as well as the rigidity of Clifford hypersurface in $S^{n}$. The $3$-dimension case of our theorems confirm two conjectures raised by Gromov in [Gro18].