\(\mathbb{R}^4\) 中限制在空间内的极小超曲面的拓扑伯恩斯坦定理
Topological Half-Space Theorems for Minimal Hypersurfaces in $\mathbb{R}^4$
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中文总结 AI 辅助
研究 \(\mathbb{R}^4\) 中极小超曲面的拓扑伯恩斯坦定理，证明曲率有界、与 \(\mathbb{R}^3\) 微分同胚且在平板中的完备恰当嵌入极小超曲面必为超平面，立方体积增长假设下半空间中的也成立。
中文摘要 AI 辅助
\(\mathbb{R}^4\) 中的三维悬链面是包含在平板中的完备嵌入极小超曲面，这表明半空间定理不能直接推广到更高维。我们证明在 \(\mathbb{R}^4\) 中这种阻碍是拓扑性的。具体而言，我们证明了一个曲率有界、与 \(\mathbb{R}^3\) 微分同胚且包含在平板中的完备、恰当嵌入极小超曲面 \(\Sigma^3\subset\mathbb{R}^4\) 必定是一个超平面。在立方体积增长的额外假设下，对于包含在半空间中的极小超曲面同样结论成立。
英文摘要
The three-dimensional catenoid in $\mathbb{R}^4$ is a complete embedded minimal hypersurface contained in a slab, showing that the half-space theorem does not extend directly to higher dimensions. We show that this obstruction is topological in $\mathbb{R}^4$. More precisely, we prove that a connected, complete embedded minimal hypersurface $Σ^3\subset\mathbb{R}^4$ contained in a half-space with bounded curvature and trivial second homology must be a hyperplane. We obtain further rigidity when a connected, complete embedded minimal hypersurface is confined to a slab. In particular, we show that if in addition the minimal hypersurface is proper and has trivial second homology then it must be a hyperplane, thus obtaining flatness without the bounded curvature assumption. If instead the minimal hypersurface has bounded curvature and finite second Betti number then it must have finite total curvature or finite index. As a corollary, we also obtain a new characterization of the three-dimensional catenoid as a minimal hypersurface confined in a slab with bounded curvature and second Betti number equal to one.
