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

Ge, Jianquan; Li, Fagui

doi:10.1090/proc/15835

A lower bound for $L_2$ length of second fundamental form on minimal hypersurfaces
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Proc. Amer. Math. Soc. 150 (2022), 2671-2684 Request permission

Abstract:

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $\delta (n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb {S}^{n+1}$, \begin{equation*} \int _{M}S \geq \delta (n)\operatorname {Vol}(M^n), \end{equation*} where $S$ is the squared length of the second fundamental form of $M^n$. The Perdomo Conjecture asserts that $\delta (n)=n$ 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 $\lambda _1(M)$ of the Laplacian involved.

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