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.