Geometric Properties Of Horizontal Sets Of Parallel Solutions Of Gaussian Curvature Flow Equations

The geometric properties of the solutions of the elliptic partial differential equations is an issue of interest for a long time in PDE. The Constant Rank Theorem is a pretty subtle statement, which usually has profound implications in geometry of solutions of PDE, especially for convexity of the solutions. About this aspect, the convexity of the level sets of the solutions is one of the the biggest concern. This paper consists of second sections. In the first section, we discuss the rank of Hessian matrix of some function of the solutions of Δu= 2 defined on 2-dimensional Riemannian manifolds is a constant. In the second section, we consider the geometric properties of the level sets of the parallel solutions of Kα-equation defined on 2-dimensional region.The main theorems are stated as follows.Theorem 1. Let Ω be a domain in S2, u∈C4(Ω) is the solution of equation Δu= 2. Let v=-(-u)1/2, suppose Hessian matrix of v is semi-positive, that is to say w=:(vij)≥0. Then the rank of the matrix in Ω is a constant.Theorem 2. Let Ω be a bounded smooth domain in R2, the v ∈ C4(Ω)∩C2(Ω) is the solution of equation Suppose|▽v|≠0 in Ω, let If (?)t, the level sets Γt are strictly convex, then is convex function.where bθθ denotes curature of curve, his the support function of...