A lower bound for the first eigenvalue of a minimal hypersurface in the sphere

Abstract

Let $\Sigma$ be a closed embedded minimal hypersurface in the unit sphere ${\mathbb{S}}^{m+1}$ and let $\Lambda ={\max }_{\Sigma }|A|$ be the norm of its second fundamental form. In this work, we prove that the first eigenvalue of the Laplacian of $\Sigma$ satisfies

and ${\lambda }_1(\Sigma )=m$ when $\Lambda \le \sqrt{m}$. In particular, this estimate improves the one obtained recently in Duncan–Sire–Spruck (2024). The proof of our main result is based on a Rayleigh quotient estimate for a harmonic extension of an eigenfunction of the Laplacian of $\Sigma$ in the spirit of Choi and Wang (1983).