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

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 \lambda_{1}(\Sigma) > \frac{m}{2}+\frac{m(m+1)}{32 (12 \Lambda+m+11)^{2}+8}, 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).