The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in R⁴, while they do not exist in positively curved closed Riemannian (n+1)-manifold when n≤5; in particular, there are no stable minimal hypersurfaces in Sⁿ⁺¹ when n≤5. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie.

中文翻译：

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稳定最小超曲面的两个刚度结果

本文的目的是证明关于完全、浸没、可定向、稳定最小超曲面的刚度的两个结果：我们证明它们在R ^{4中是超平面，而在正曲闭黎曼 (} n +1)-中不存在。n ≤ 5时的流形；特别是，当n ≤ 5 时， S ^{n +1}中不存在稳定的最小超曲面。第一个结果最近也被 Chodosh 和 Li 证明，第二个结果是关于具有有限指数的最小曲面的更一般结果的结果。这两个定理都依赖于保形方法，其灵感来自 Fischer-Colbrie 的经典著作。