Equi-centro-affine extremal hypersurfaces in ellipsoid

TL;DR

This paper investigates special extremal hypersurfaces in ellipsoids, deriving variational formulas, analyzing stability, classifying extremal hypersurfaces and curves, and establishing an isoperimetric inequality in the context of centro-affine geometry.

Contribution

It provides the first and second variational formulas for invariant areas, classifies extremal hypersurfaces and curves, and introduces a new isoperimetric inequality in equi-centro-affine geometry.

Findings

Circles of radius √6/3 on S²(1) are equi-centro-affine maximal.

Complete classification of compact isoparametric extremal hypersurfaces.

Identification of a family of transcendental extremal curves x_{p,q}.

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 ${}^{ec}\hspace{-1mm}L^3\leq (4\pi-A)(2\pi-A)A$ on $\mathbb{S}^2(1)$.