Reduces CMC hypersurfaces with double horocyclic symmetry in H²×H² to an autonomous ODE, solves explicitly in three regimes, proves existence/uniqueness, and classifies equilibria as H³, H²×R, Sol₃ and semidirect-product metrics.
Hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant sectional curvature
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abstract
In this paper, we classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant sectional curvature. In contrast to $\mathbb{S}^2\times\mathbb{S}^2$, the resulting examples for $\mathbb{H}^2\times\mathbb{H}^2$ exhibit more diversity, and we construct a special example with non-constant product angle function. For $\mathbb{S}^2\times\mathbb{S}^2$, however, the product angle function of any constant sectional curvature hypersurface is identically zero. As a byproduct, we classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant product angle function and constant mean curvature (or constant scalar curvature).
fields
math.DG 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Constant mean curvature hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$ with double horocyclic symmetry
Reduces CMC hypersurfaces with double horocyclic symmetry in H²×H² to an autonomous ODE, solves explicitly in three regimes, proves existence/uniqueness, and classifies equilibria as H³, H²×R, Sol₃ and semidirect-product metrics.