Equi-centro-affine extremal hypersurfaces in ellipsoid
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916590247739392 |
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| author | Yang, Yun Qu, Changzheng |
| author_facet | Yang, Yun Qu, Changzheng |
| contents | 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 $\mathbb{S}^2(1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equi-centro-affine extremal hypersurfaces in ellipsoid Yang, Yun Qu, Changzheng Differential Geometry 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 $\mathbb{S}^2(1)$. |
| title | Equi-centro-affine extremal hypersurfaces in ellipsoid |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2501.18127 |