Do Carmo's problem for CMC hypersurfaces in ${\mathbb{R}}^6$

In this paper, we prove that complete noncompact constant mean curvature hypersurfaces in ${\mathbb{R}}^6$ with finite index must be minimal. This provides a positive answer to do Carmo’s question in dimension $6$. The proof strategy is also applicable to ${\mathbb{R}}^4$ and ${\mathbb{R}}^5$, thereby providing alternative proofs for those previously resolved cases.