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

From papers

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.

Progress summary

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Equivalent formulations 1

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).

Sources & referencesView supporting material

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.

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Progress summary

Equivalent formulations 1

The Twin-Bubble Conjecture for constant mean curvature hypersurfaces

Primary source

Additional references