Geometric Properties Of The Slower Line On The Minimum Surface Of Two - Dimensional Riemannian Manifold

In this thesis.we discuss some geometrical properties of convex level sets of min-imal graph on 2-dimensional Riemannian manifolds.For the minimal graph defined on 2-dimensional Rimannian manifolds,We consider the steepest descent curvature that satisfied some differential inequalities,and use the maximum principle to prove the geo-metrical properties.The thesis consists of three chapters.In chapter 1,we briefly recall the history of convex level sets of solutions to elliptie equations,and some development about it. At the same timemwe also introduce the main conclusion. In chapter 2,we list the notations and preliminaries being used during the following proof.In chapter 3, we set out to prove the main theorems.Theorem 1.3:Let Ω be a smooth bounded domain on the two-dimensional Rie-mannian manifolds M2 with constant Gaussian curvature denoted by K.Let u∈C4(Ω) ∩C2(Ω)be the minimal graph defined on Ω.Assume |▽u|≠0,Let J be the curva-ture of the steepest descent of u.For K≤0,the function(|▽u|2/1+|▽u|2)-1/2 J attains its maximum on the boundary (?)Ω.