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Minimal hypersurfaces in the product of two spheres with index one
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-06-29 , DOI: 10.1007/s00526-021-02007-4
Hang Chen

Given \(n_1\ge n_2\ge 2\), let \(\Sigma \) be an orientable, minimal hypersurface of \(\mathbb {S}^{n_1}(1)\times \mathbb {S}^{n_2}(a)\) with index one. Assume \(\Sigma \) is closed and either smooth or singular with a singular set \({{\,\mathrm{sing}\,}}(\Sigma )\) satisfying \(\mathcal {H}^{n-2}({{\,\mathrm{sing}\,}}(\Sigma ))=0\). By using the almost product structure, we prove that, when \(a^{2}\ge \frac{n_2}{n_1-1}\) or \(a^{2}\le \frac{n_2-1}{n_1}\), such \(\Sigma \) must be totally geodesic. As an application, combining with the results of X. Zhou [37], we compute the width of \(\mathbb {S}^{n_1}(1)\times \mathbb {S}^{n_2}(a)\).



中文翻译:

索引为 1 的两个球体的乘积中的最小超曲面

给定\(n_1\ge n_2\ge 2\),令\(\Sigma \)\(\mathbb {S}^{n_1}(1)\times \mathbb {S}^{ n_2}(a)\)索引为 1。假设\(\Sigma \)是封闭的并且是平滑的或奇异的,具有满足\(\mathcal {H}^{n的奇异集\({{\,\mathrm{sing}\,}}(\Sigma )\) -2}({{\,\mathrm{sing}\,}}(\Sigma ))=0\)。通过使用几乎乘积结构,我们证明,当\(a^{2}\ge \frac{n_2}{n_1-1}\)\(a^{2}\le \frac{n_2-1} {n_1}\),这样的\(\Sigma \)必须是完全测地线。作为一个应用,结合 X. Zhou [37] 的结果,我们计算了\(\mathbb {S}^{n_1}(1)\times \mathbb {S}^{n_2}(a)\)

更新日期:2021-06-29
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