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We prove that any complete immersed two-sided mean convex translating soliton Σ ⊂ R 3 for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in R 3 is the axisymmetric "bowl soliton". We also show that if the mean curvature of Σ tends to zero at infinity, then Σ can be represented as an entire graph and so is the "bowl soliton". Finally we classify the asymptotic behavior of all locally strictly convex graphical translating solitons defined over strip regions.

Abstract. In [7], Guan, Ren and Wang obtained a C 2 a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation σ 2 (κ(X)) = f (X, ν(X)). In this note, we give a simpler proof of this result, and extend it to space forms.

Abstract. In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to n-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

Abstract. We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H n+1 satisfying f (κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if Γ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a "global interior curvature estimate".