Equi-centro-affine extremal hypersurfaces in ellipsoid

TL;DR

The paper develops an in-depth equi-centro-affine variational framework for hypersurfaces in ellipsoids, deriving the first and second variation formulas for the invariant area $\Sigma_{eca}$ under unimodular centro-affine flows and identifying the Euler-type extremality condition ${}^{ec}H=0$. It then analyzes the stability of equi-centro-affine arc-length on $\mathbb{S}^2$, proving that the circle of radius $r=\sqrt{6}/3$ is the unique embedded maximizer under area constraints and establishing an isoperimetric-type bound ${}^{ec}L^3\le (4\pi-A)(2\pi-A)A$ on $\mathbb{S}^2(1)$. The work further classifies compact isoparametric equi-centro-affine extremal hypersurfaces on $\mathbb{S}^{n+1}$, including Clifford-type products, and analyzes closed equi-centro-affine extremal curves on the sphere, showing they belong either to planar circles or to the $\mathbf{x}_{p,q}$ family with restricted $p/q$ intervals. Collectively, these results deepen the understanding of affine-invariant extremals and the geometric-analytic structure of equi-centro-affine flows on spherical and ellipsoidal backgrounds, and they establish a sharp equi-centro-affine isoperimetric inequality on $\mathbb{S}^2(1)$.

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

Figures (4)

• Figure 1: The ranges for $C_1$ and $C_2$ determined by Proposition \ref{['prop-bd']}.
• Figure 2: The incremental or decremental processes of $\Lambda^{\Theta}$.
• Figure 3: The closed extremal curves on unit sphere.
• Figure 4: The comparison of different method to calculate $\Lambda^{\Theta}$.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof