Compact Surfaces with Constant Gaussian Curvature in Product Spaces

Abstract

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 complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \({\mathbb{H}^{2}\times\mathbb{R}}\) and positive constant Gaussian curvature greater than 1 in \({\mathbb{S}^{2}\times\mathbb{R}}\) whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.