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 $H^2>1$ 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.