Optimal upper threshold for strongly stable CMC hypersurfaces

Determine the least positive number H_*^n such that no complete strongly stable constant-mean-curvature hypersurface immersed in hyperbolic space \mathbb{H}^n exists with mean curvature H>H_*^n, and determine whether this optimal threshold is attained by a CMC hypersurface with distinguished properties.

Background

The paper constructs complete, properly embedded, strictly strongly stable CMC hypersurfaces in \mathbb{H}n for every n\geq 4 with mean curvature in a neighborhood of the critical value n-1. These examples have infinitely many ends and demonstrate that several previously considered rigidity statements fail, including rigidity for mean curvature above n-1 in the relevant dimensions.

The authors therefore ask for the sharp global upper threshold beyond which complete strongly stable CMC hypersurfaces cannot exist. They also ask whether the threshold is realized by an especially characterized hypersurface. In dimension three, the paper records that the threshold is H_*3=2 and that equality is attained uniquely by horospheres, providing the known model case for the unresolved higher-dimensional problem.

References

In light of Theorem \ref{thm:main}, it is natural to ask: what is the least positive number $H_n$ such that there exists no complete strongly stable CMC hypersurface immersed in $\mathbb{H}n$ with mean curvature $H>H_n$? Is the optimal value attained by some CMC hypersurface with distinguished properties? Note that $H_*3 = 2$, and equality is attained uniquely by horospheres in this case .

— Stable Constant Mean Curvature Hypersurfaces in $\mathbb{H}^n$  (2609.34283 - Gaia et al., 28 Sep 2026) in Section 1, subsection “Related literature”