Splitting and pinching for complete convex surfaces in \(E^3\) (Q1380853)

Let \( M \) be a complete noncompact \( C^2 \)-surface immersed in Euclidean space \( \mathbb{R}^3 \) with Gauss curvature \( K \geq 0 \). Assume further that the mean curvature \( H \) satisfies \( \lambda \leq H \leq 2 \lambda \) for some constant \( \lambda > 0 \). The author shows that \( M \) must be a cylinder and, moreover, that the coefficient \( 2 \) is optimal, thus verifying a conjecture of \textit{P. Kohlmann} in this situation [Geom. Dedicata 60, 125-143 (1996; Zbl 0849.52005)]. The proof uses a graph representation of \( M \) over a domain in \( \mathbb{R}^2 \) and Crofton's formula in order to estimate the width of the projection to \( \mathbb{R}^2 \). Another ingredient of the proof are related integral formulas for \( H \) and \( K \).