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The aim of this paper is to prove two results concerning the rigidity ofcomplete, immersed, orientable, stable minimal hypersurfaces: we show that theyare hyperplane in $\mathbb{R}^4$, while they do not exist in positively curvedclosed Riemannian $(n+1)$-manifold when $n\leq 5$; in particular, there are nostable minimal hypersurfaces in $\mathbb{S}^{n+1}$ when $n\leq 5$. The firstresult was recently proved also by Chodosh and Li, and the second is aconsequence of a more general result concerning minimal surfaces with finiteindex. Both theorems rely on a conformal method, inspired by a classical workof Fischer-Colbrie.

Two rigidity results for stable minimal hypersurfaces