A lower bound for the first eigenvalue of a minimal hypersurface in the sphere
Let $Σ$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $Λ=\max\limits_Σ|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the Laplacian of $Σ$ satisfies $$λ_1(Σ)> \dfrac{m}{2}+\frac{m(m+1)}{32(12Λ+m+11)^2+8},$$ and $λ_1(Σ)=m$, when $Λ\le\sqrt{m}$. 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 $Σ$ in the spirit of \cite{choi1983first}.
math.DG↗