A characterization of the grim reaper cylinder

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

Translating solitons foliated by spheres

In this paper, we consider translating solitons in [Formula: see text] which is foliated by spheres. In three-dimensional Euclidean space, we show that such a translating soliton is a surface of revolution and the axis of revolution is parallel to the translating direction of the translating soliton. We also show that the same result holds for a higher dimension case with a hypersurface foliated by spheres in parallel hyperplanes that are perpendicular to the translating direction.

Separation of variables in equations of mean-curvature type

We obtain all the solutions of types u(x, y) = f(x) + g(y) and u(x, y) = f(x)g(y) for three known mean-curvature-prescribed equations, namely, the capillary equation, the translating soliton equation and the two-dimensional analogue of the catenary.

On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space II

Let u be a smooth convex function in ℝn and the graph M∇u of ∇u be a space-like translating soliton in pseudo-Euclidean space [Formula: see text] with a translating vector [Formula: see text], then the function u satisfies [Formula: see text] where ai, bi and c are constants. The Bernstein type results are obtained in the course of the arguments.

TRANSLATING SOLITONS TO SYMPLECTIC AND LAGRANGIAN MEAN CURVATURE FLOWS

In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the Kähler angle α of a symplectic translating soliton with max |A| = 1 satisfies that [Formula: see text] where T is the direction in which the surface translates.