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We use a phase space analysis to give some classification results for rotational hypersurfaces in R n+1 whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in S n , we show that a Delaunay-type classification holds for this class of hypersurfaces. We also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed (non-constant) mean curvature is much richer than in the constant mean curvature case.Mathematics Subject Classification: 53A10, 53C42, 34C05, 34C40 arXiv:1902.09405v1 [math.DG]

We develop a global theory for complete hypersurfaces in R n+1 whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in R n+1 , and also that of self-translating solitons of the mean curvature flow. For the particular case n = 2, we will obtain results regarding a priori height and curvature estimates, non-existence of complete stable surfaces, and classification of properly embedded surfaces with at most one end. Basic properties of H-hypersurfaces

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If ξ is a Killing vector field of the hyperbolic space H 3 whose flow are parabolic isometries, a surface Σ ⊂ H 3 is a ξ-translator if its mean curvature H satisfies H = ⟨N, ξ⟩, where N is the unit normal of Σ. We classify all ξ-translators invariant by a one-parameter group of rotations of H 3 , exhibiting the existence of a new family of grim reapers. We use these grim reapers to prove the non-existence of closed ξ-translators.