Some Geometric Properties Of Minimal Graphs In Space With Non-positive Curvature

For the minimal graph defined on 2-dimensional space forms with nonpositive cur-vature,we derive some geometrical properties,including the regularity and the strict convexity of the level sets,the curvature description of its steepest descents.The paper is organized as follows:in section 2,we list the notations and the preliminaries being used during the process of the proof,especially,we deduce the equation of the minimal graph on Riemannian manifolds and the curvature of the level sets and the steepest descent.In section 3,we come to discuss the strict convexity of the level sets of the minimal graph defined on space forms with nonpositive curvature.In section 4,we get a theorem of curvature estimate of the steepest descents of the minimal graph and make strict proof of this theorem.The main results is as follows:Thm 1 Let(M2,g)be a space form with constant sectional curvature ??0,and?0 and ?1 be bounded smooth strictly convex domains in M2,(?)1(?)?0.Assume that the following minimal graph equation defined on ? = ?0\(?)1 (?) has a smooth solution u.Then ?u?0 is valid everywhere in ? and all the level sets of u are strictly convex with respect to ?u.Corollary 1 Let(M2,g)be a space form with constant sectional curvature ? ? 0,and ?0 and ?1 be bounded smooth strictly convex domains in M2 and (?)1(?)?0.Suppose that ua is the solution to(1.1).Then |?u| increases strictly along the gradient direction.Thm 2 Let u be the solution to equation(1.1)on a domain in Riemannian manifold M2 with nonpositive constant Gaussian curvature denoted by ?.Let J be the curvature of the steepest descent of u.For ? = |?u|-1 J,the following relation is satisfied:(?)Here,aij=(1+|?u|2)?ij-uiuj is the minimal surface operator.