A recent paper proves that any complete, non‑compact, two‑sided, properly embedded hypersurface in Euclidean space with constant mean curvature and finite Morse index is necessarily minimal. The result holds in every dimension and does not rely on curvature or volume bounds.

The proof builds a weight function μ equal to the reciprocal of the radius of the largest tangent ball lying in the region toward which the surface’s mean curvature vector points. μ satisfies the differential inequality Δμ+|A|²μ≥nHμ², and a weighted stability estimate nH∫μ³φ²≤∫μ²|∇φ|² for all compactly supported φ.

These estimates allow the authors to rule out non‐zero mean curvature when the Morse index is finite.

Proper embeddedness is essential; without it the construction of the weighted measure ρdV and the resulting differential inequality cannot be carried out.

The tangent‑ball argument gives a polynomial upper bound on the weighted volume of the hypersurface inside any Euclidean ball: ∫_{Σ∩B_R}ρ≤C(R+H−1)^{n+1}. This bound is used to control the growth of μ and to establish the stability estimate above.

The work settles the properly embedded case of a longstanding question posed by do Carmo regarding finite‑index surfaces.

See arXiv:2611.08454v1 for the full proof and technical details.