A lower bound for $L_2$ length of second fundamental form on minimal hypersurfaces

TL;DR

This paper establishes a lower bound for the integral of the squared second fundamental form on minimal hypersurfaces in spheres, advancing understanding of geometric inequalities related to the Perdomo Conjecture.

Contribution

It proves a weak version of the Perdomo Conjecture, providing a positive lower bound depending only on dimension for the integral of the second fundamental form.

Findings

Established a lower bound for the integral of the squared second fundamental form.

Derived new integral inequalities for minimal hypersurfaces.

Obtained Simons-type pinching results involving the first eigenvalue of the Laplacian.

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 $\mathbb{S}^{n+1}$, $$\int_{M}S \geq \delta(n){\rm Vol}(M^n),$$ 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.