{"ID":22994680,"CreatedAt":"2026-09-17T04:01:24.020413181Z","UpdatedAt":"2026-09-17T04:01:24.020413181Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19120","arxiv_id":"2609.19120","title":"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 $2\\le n\\le6$ 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 $H^2\u003e1$ for surfaces to $H^2\\ge5/3$ 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.","short_abstract":"We study complete two-sided constant-mean-curvature hypersurfaces of dimensions $2\\le n\\le6$ 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 und...","url_abs":"https://arxiv.org/abs/2609.19120","url_pdf":"https://arxiv.org/pdf/2609.19120v1","authors":"[\"Zihao Wang\"]","published":"2026-09-16T17:43:26Z","proceeding":"math.DG","tasks":"[\"math.DG\"]","methods":"[]","has_code":false}