Optimal constants of $L^2$ inequalities for closed nearly umbilical hypersurfaces in space forms

TL;DR

This paper proves the optimality of constants in $L^2$ inequalities that measure the stability of closed hypersurfaces with non-negative Ricci curvature in space forms, extending previous stability results.

Contribution

It establishes the sharpness of constants in existing $L^2$ inequalities for nearly umbilical hypersurfaces in space forms.

Findings

Constants in $L^2$ inequalities are proven to be optimal.

Extends stability results to more general hypersurfaces.

Provides precise bounds for geometric inequalities.

Abstract

Let $\Sigma$ be a smooth closed hypersurface with non-negative Ricci curvature, isometrically immersed in a space form. It has been proved in \cite{P}, \cite{CZ}, and \cite{C2} that there are some $L^2$ inequalities on $\Sigma$ which measure the stability of closed umbilical hypersurfaces or more generally, closed hypersurfaces with traceless Newton transformation of the second fundamental form. In this paper, we prove that the constants in these inequalities are optimal.