An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Sphere

Abstract Suppose that ^n S^n+1 is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue ₁ of the induced Laplace–Beltrami operator on satisfies ₁ n2+ a₍ (^6 + b₍) ^-1, where a₍ and b₍ are explicit dimensional constants and is an upper bound for the length of the second fundamental form of. This provides the first explicitly computable improvement on Choi and Wang’s lower bound ₁ n2 without any further assumptions on.