Some results on Chern's problem

TL;DR

This paper proves a pinching result for compact minimal hypersurfaces in spheres, showing that if the squared length of the second fundamental form is close to a specific value, then the hypersurface must be a Clifford minimal hypersurface.

Contribution

It establishes a new pinching constant depending only on the dimension, confirming a conjecture related to Chern's problem for minimal hypersurfaces.

Findings

For dimensions n ≥ 6, the pinching constant δ(n) = n/23.

Hypersurfaces with S in [n, n + δ(n)] are Clifford minimal hypersurfaces.

Confirmed a specific case of Chern's problem for minimal hypersurfaces.

Abstract

For a compact minimal hypersurface $M$ in $S^{n+1}$ with the squared length of the second fundamental form $S$ we confirm that there exists a positive constant $\de(n)$ depending only on $n,$ such that if $n\leq S\leq n +\delta(n)$, then $S\equiv n$, i.e., $M$ is a Clifford minimal hypersurface, in particular, when $n\ge 6,$ the pinching constant $\de(n)=\f{n}{23}.$