Abstract. In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space.

We investigate complete minimal hypersurfaces in the Euclidean space R 4 , with Gauss-Kronecker curvature identically zero. We prove that, if f : M 3 → R 4 is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then f (M 3 ) splits as a Euclidean product L 2 × R, where L 2 is a complete minimal surface in R 3 with Gaussian curvature bounded from below.2000 Mathematics Subject Classification. 53C42.

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Abstract. In this article we prove that a connected and properly embedded translating soliton in R 3 with uniformly bounded genus on compact sets which is C 1 -asymptotic to two planes outside a cylinder, either is flat or coincides with the grim reaper cylinder.

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