A Geometric Property Of Parallel Solutions Of Mean Curvature Flow

The study of the geometric properties of the elliptic partial differential equations is an important subject, especially, people is interested in the curvature estimates in the convex level sets of the elliptic partial differential equations. This paper introduces a special mean curvature type equation defined on 2-dimensional, the curvature estimates in the convex level sets of its. And we use the maximum principle to complete this part-Theorem 1.1 Let Ω(?)R2, u ∈C4(Q) ∩ C2(Ω be a solution of the mean curvature type equation in Ω, i.e. Assume |(?)i|≠ 0 in Ω, and if its respects to normal (?)u the level sets of u are strictly convex. Then the function |(?)u|-2K attains it minimum on the boundary, there let K be the curvature of the level sets.This paper also gives Liouville type theorems of the harmonic functions who satisfy the sublinear growth in the whole space Hn.Theorem 1.2 Assume u is a harmonic function of the Heisenberg group Hn, and satisfying the sublinear growth then u is a constant.There r(ζ) representative the distance from ζ to the original point.