Gehring link problem, focal radius and over-torical width

abstract: In this note, we study the Gehring link problem in the round sphere, which motivates our study of the width of a band in positively curved manifolds. Using the same idea, we are able to prove a sphere theorem for hypersurface in the round $\SS^n$ provided that its normal injectivity radius is large. A rigidity theorem for Clifford hypersurfaces in $\SS^n$ is also proved. The $3$-dimensional case of our theorems confirm two conjectures raised by Gromov (2018).