Complete constant mean curvature surfaces and bernstein type theorems in 𝕄2×ℝ

Abstract

In this paper we study constant mean curvature surfaces Σ in a product space, 𝕄2×ℝ, where 𝕄2 is a complete Riemannian manifold. We assume the angle function ν = (N,∂/∂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 𝕄2×(t), or 𝕄2≡ℝ2with 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 M2 of constant geodesic curvature 2H. This result is optimal. We also prove a non-existence result concerning complete multigraphs in 𝕄2×ℝ, when c(𝕄2) < 0. © 2009 J. differential geometry.