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 $\Lambda=\max\limits_{\Sigma}|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the Laplacian of $\Sigma$ satisfies $$\lambda_1(\Sigma)> \dfrac{m}{2}+\frac{m(m+1)}{32(12\Lambda+m+11)^2+8},$$ and $\lambda_1(\Sigma)=m$, when $\Lambda\le\sqrt{m}$. In particular, this estimate improves the one obtained recently in \cite{duncan2023improved}. 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 \cite{choi1983first}.