Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature

TL;DR

This paper proves a lower bound on the second fundamental form's squared norm for certain minimal hypersurfaces in spheres, providing an approximate validation of the Chern conjecture related to scalar curvature gaps.

Contribution

It establishes a new inequality for the second fundamental form of minimal Willmore hypersurfaces with constant scalar curvature, advancing understanding of the Chern conjecture.

Findings

Established a lower bound for S in terms of n.

Provided partial verification of the Chern conjecture.

Linked scalar curvature conditions to geometric inequalities.

Abstract

In this paper, we prove that for an $n$-dimensional closed minimal Willmore hypersurface $M^n$ with constant scalar curvature in the unit sphere $\mathbb{S}^{n+1}$, the squared norm $S$ of the second fundamental form of $M^n$ satisfies $S\geqslant n+\frac{4n+9-\sqrt{4 n^{2}+60 n+81}}{2}$ if $S>n$. This proves, in the approximate sense, the Chern conjecture about the second gap ($S\geqslant 2n$ if $S>n$), which will be fully verified under a further inequality condition about the 4-th mean curvature.