A lower bound for 𝐿₂ length of second fundamental form on minimal hypersurfaces

Abstract

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant δ ( n ) > 0 \delta (n)>0 depending only on n n such that on any closed embedded, non-totally geodesic, minimal hypersurface M n M^n in S n + 1 \mathbb {S}^{n+1} , ∫ M S ≥ δ ( n ) Vol ⁡ ( M n ) , \begin{equation*} \int _{M}S \geq \delta (n)\operatorname {Vol}(M^n), \end{equation*} where S S is the squared length of the second fundamental form of M n M^n . The Perdomo Conjecture asserts that δ ( n ) = n \delta (n)=n which is still open in general. As byproducts, we also obtain some integral inequalities and Simons-type pinching results on closed embedded (or immersed) minimal hypersurfaces, with the first positive eigenvalue λ 1 ( M ) \lambda _1(M) of the Laplacian involved.