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

Jingche Chen; Han Hong; Haizhong Li

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

Differential Geometry 2025-04-24 v2

Authors: Jingche Chen , Han Hong , Haizhong Li

Abstract

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.

Keywords

hypersurfaces minimal hypersurfaces and morse index

Cite

@article{arxiv.2503.08107,
  title  = {Do Carmo's problem for CMC hypersurfaces in $\mathbb{R}^6$},
  author = {Jingche Chen and Han Hong and Haizhong Li},
  journal= {arXiv preprint arXiv:2503.08107},
  year   = {2025}
}

Comments

Some modifications to the manuscript : 1, simply the proof without using the conformal change by $1/r^2$; 2, correct the proof of the main theorem and provide more details to the volume estimate of geodesic balls

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

Abstract

Keywords

Cite

Comments

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