Let (M^n+1,g) be a closed Riemannian manifold with 3≤ n+1≤ 7, and let c>0. A closed hypersurface has constant mean curvature c if its mean curvature is equal to c at every point. Twin Bubble Conjecture. Every closed Riemannian manifold (M^n+1,g) with 3≤ n+1≤ 7 admits at least two distinct closed hypersurfaces with constant mean curvature c for every c>0. This is presented as a higher-dimensional analogue of Arnold's conjecture. The source cites partial results, but does not state that the conjecture has been resolved.

Twin Bubble Conjecture for constant-mean-curvature hypersurfaces

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Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\le n+1\le 7$ , and let $c>0$ . A closed hypersurface has constant mean curvature $c$ if its mean curvature is equal to $c$ at every point. Twin Bubble Conjecture. Every closed Riemannian manifold $(M^{n+1},g)$ with $3\le n+1\le 7$ admits at least two distinct closed hypersurfaces with constant mean curvature $c$ for every $c>0$ . This is presented as a higher-dimensional analogue of Arnold's conjecture. The source cites partial results, but does not state that the conjecture has been resolved.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Twin-Bubble Conjecture for constant mean curvature hypersurfaces
Let $M^{n+1}$ be a closed Riemannian manifold of dimension $3\le n+1\le 7$ . An almost-embedded hypersurface is a hypersurface with the regularity and self-intersection behavior intended by the source.

Twin-Bubble Conjecture. For every $c>0$ , there are at least two almost-embedded hypersurfaces in $M$ with constant mean curvature $c$ .

This conjecture predicts multiplicity of constant mean curvature hypersurfaces in dimensions where the relevant regularity theory applies. The source states that related conjectures remain open.
source: Liam Mazurowski and Xin Zhou, “Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature”, arXiv:2511.08228 (2025).

References

Primary source

Talant Talipov, “Equivalence of Almgren-Pitts and phase-transition half-volume spectra”, arXiv:2604.01091 (2026).

Additional references

3 papers in this index state this conjecture (2024–2026). The statement above is taken from the most recent of them; the others are arXiv:2408.13864, arXiv:2405.00595.

Posted 1 month ago

Twin Bubble Conjecture for constant-mean-curvature hypersurfaces

The Twin-Bubble Conjecture for constant mean curvature hypersurfaces

Primary source

Additional references

Progress summary

Solutions 0