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We prove that the only compact surfaces of positive constant Gaussian curvature in $${\mathbb{H}^{2}\times\mathbb{R}}$$ (resp. positive constant Gaussian curvature greater than 1 in $${\mathbb{S}^{2}\times\mathbb{R}}$$ ) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational...
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