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On the structure of stable constant anisotropic mean curvature hypersurfaces
Abstract
We prove that a stable noncompact hypersurface with constant anisotropic mean curvature in ℝn+1 has only one end for n ≤ 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 ℝ7 (n=6), in the case of anisotropic minimal hypersurfaces. The proof relies on the existence of a Sobolev-type inequality.
Related
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- Multiplicity One for Strictly Stable Anisotropic Allen-Cahn Minimal Hypersurfaces
- Hypersurfaces of constant higher order mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity
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BibTeX
@misc{pontuale2026structure,
title = {{On the structure of stable constant anisotropic mean curvature hypersurfaces}},
author = {Claudia Pontuale},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27480},
eprint = {2609.27480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27480},
}
RIS
TY - UNPB
TI - On the structure of stable constant anisotropic mean curvature hypersurfaces
AU - Claudia Pontuale
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27480
UR - https://arxiv.org/abs/2609.27480
AB - We prove that a stable noncompact hypersurface with constant anisotropic mean curvature in ℝ^n+1 has only one end for n ≤ 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 ℝ^7 (n=6), in the case of anisotropic minimal hypersurfaces. The proof relies on the existence of a Sobolev-type inequality.
ER -