2008/08/25 by José M. Espinar, Espinar, Jose M., Harold Rosenberg +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.0808.3412
openalex publication_date 2008/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study constant mean curvature surfaces Σ in a product space, \mathbbM2× ℝ, where \mathbbM2 is a complete Riemannian manifold. We assume the angle function ν= \metaN∂t does not change sign on Σ. We classify these surfaces according to the infimum c(Σ) of the Gaussian curvature of the projection of Σ. When H ≠ 0 and c(Σ)≥ 0, then Σ is a cylinder over a complete curve with curvature 2H. If H=0 and c(Σ) ≥ 0, then Σ must be a vertical plane or Σ is a slice \mathbbM2 × t, or \mathbbM2 ≡ ℝ2 with the flat metric and Σ is a tilted plane (after possibly passing to a covering space). When c(Σ)<0 and H>√(-c(Σ)) /2, then Σ is a vertical cylinder over a complete curve of \mathbbM2 of constant geodesic curvature 2H. This result is optimal. We also prove a non-existence result concerning complete multi-graphs in \mathbbM2× ℝ, when c(\mathbbM2)<0.