Equi-centro-affine extremal hypersurfaces in ellipsoid

Abstract

This paper explores equi-centro-affine extremal hypersurfaces in an ellipsoid. By analyzing the evolution of invariant submanifold flows under centro-affine unimodular transformations, we derive the first and second variational formulas for the associated invariant area. Stability analysis reveals that the circles with radius $r=\sqrt{6}/3$ on ${\mathbb{S}}^2(1)$ are characterized as being equi-centro-affine maximal. Furthermore, we provide a detailed classification of the compact isoparametric equi-centro-affine extremal hypersurfaces on $(n+1)$-dimensional sphere, as well as the generalized closed equi-centro-affine extremal curves on $2$-dimensional sphere. These curves are shown to belong to a family of transcendental curves ${\mathrm{x}}_{p,q}$ ($p,q$ are two coprime positive integers satisfying that $1/2<p/q<1$ ). Additionally, we establish an equi-centro-affine version of isoperimetric inequality ${}^{ec}{L}^3\le (4\ddot{I}€-A)(2\ddot{I}€-A)A$ on ${\mathbb{S}}^2(1)$.