Finite index constant mean curvature hypersurfaces in space forms of dimension at most seven
Abstract
We study complete two-sided constant-mean-curvature hypersurfaces of dimensions 2≤ n≤6 and finite Morse index in simply connected space forms. In the round sphere the immersed domain is compact; in Euclidean space every noncompact example is minimal; in hyperbolic space of curvature −1 we obtain compactness under explicit dimension-dependent mean-curvature thresholds, from H2>1 for surfaces to H2≥5/3 in dimension six. The conclusions also hold for the volume-constrained index, without properness, volume-growth, or curvature-bound assumptions. A common Green-function argument uses explicit rational combinations of eight identities and the stable Bernstein theorem of Hong-Li-Wang. Stable hyperbolic tubes show that no mean-curvature threshold independent of dimension can give compactness in all dimensions.
Related
- Hypersurfaces with constant sectional curvature of a product of two space forms
- Stable Constant Mean Curvature Hypersurfaces in ℍn
- Hypersurfaces of constant higher order mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity
- On the structure of stable constant anisotropic mean curvature hypersurfaces
- Sphere Theorems for Immersed Hypersurfaces with Constant Scalar Curvature
BibTeX
@misc{wang2026finite,
title = {{Finite index constant mean curvature hypersurfaces in space forms of dimension at most seven}},
author = {Zihao Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19120},
eprint = {2609.19120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19120},
}
RIS
TY - UNPB
TI - Finite index constant mean curvature hypersurfaces in space forms of dimension at most seven
AU - Zihao Wang
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19120
UR - https://arxiv.org/abs/2609.19120
AB - We study complete two-sided constant-mean-curvature hypersurfaces of dimensions 2≤ n≤6 and finite Morse index in simply connected space forms. In the round sphere the immersed domain is compact; in Euclidean space every noncompact example is minimal; in hyperbolic space of curvature −1 we obtain compactness under explicit dimension-dependent mean-curvature thresholds, from H^2>1 for surfaces to H^2≥5/3 in dimension six. The conclusions also hold for the volume-constrained index, without properness, volume-growth, or curvature-bound assumptions. A common Green-function argument uses explicit rational combinations of eight identities and the stable Bernstein theorem of Hong-Li-Wang. Stable hyperbolic tubes show that no mean-curvature threshold independent of dimension can give compactness in all dimensions.
ER -