Geometric Properties Of Maximal Space-like Hypersurfaces On Two-dimensional Riemannian Manifolds

The elliptic partial differential equation plays an extremely indispensable role in math-ematics,as well as physics etc.The equation of strictly maximal space-like hypersurfaces is one of the representative subject.In this paper,we research the strictly maximal space-like hypersurfaces equation on 2-dimensional Riemannian mainfolds,consider the convexity of the level sets and show the curvature estimate of the steepest descent about the solution of equation.There are four parts in this paper: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 Maximal Space-like graph on Riemannian manifolds and the curvature of the level sets and the steepest descent,constant rank theorem about the level curves,maximum principle.In section 3,we will derive the strict convexity of the level sets.In section 4,we come to the curvature estimate of the steepest descents of the maximal space-like hypersurface.The following are the main results.Theorem 1.Let(M2,g)be a space form with constant sectional curvature ∈,andΩ0 and Ω1 be bounded smooth strictly convex domains in M2,(?)1(?)Ω0.Assume that the following maximal strictly space-like graph equation defined on Ω=Ω0\(?)1 (?)(1.1)has a smooth strictly space-like solution u on (?).Then ▽u ≠ 0 is valid everywhere on (?)and all the level sets of u are strictly convex with respect to ▽u.Theorem 2.Let u be the solution to equation(1.1)on a domain in Riemannian manifold M2 with constant Gaussian curvatture denoted by ∈.Let J be the curvature of the steepest descent of u.For φ=|▽u|-1J,the following relation is satisfied:(?)Here,aij =(1-|▽u|2)δij+uiuj.