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arXiv2026

On the structure of stable constant anisotropic mean curvature hypersurfaces

We prove that a stable noncompact hypersurface with constant anisotropic mean curvature in $\mathbb{R}^{n+1}$ has only one end for $n \le 6$, under a natural ellipticity condition on the anisotropy. This provides an anisotropic counterpart of the one-end theorem for stable constant mean curvature hypersurfaces. We remark that the result is also new in $\mathbb R^7 (n=6)$, in the case of anisotropic minimal hypersurfaces. The proof relies on the existence of a Sobolev-type inequality.

math.DG↗

arXiv2026

Finite Index and Do Carmo Question for Constant Mean Curvature Hypersurfaces

We prove that any finite $δ$-index hypersurface $M$ in ${\mathbb R}^{n+1}$ with constant mean curvature must be minimal, provided either of the following conditions holds: - the volume growth of $M$ is sub-exponential; - the Ricci curvature of $M$ satisfies $\operatorname{Ric}_M\geq -\frac{3(1-δ)}{n-1}|A|^2g,$ where $A$ is the second fundamental form and $g$ is the metric on $M.$ In the second case, our result further implies that, in addition to being minimal, such an $M$ must be a hyperplane. We emphasize that no restriction on the dimension is imposed. Moreover, the statement in the second case is new even for finite index hypersurfaces ($δ=0$).

math.DG↗